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Theorem csbfv12gALTVD 39650
Description: Virtual deduction proof of csbfv12 6370. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbfv12 6370 is csbfv12gALTVD 39650 without virtual deductions and was automatically derived from csbfv12gALTVD 39650.
1:: (   𝐴𝐶   ▶   𝐴𝐶   )
2:1: (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦} = { 𝑦}   )
3:1: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹 “ {𝐵 }) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵})   )
4:1: (   𝐴𝐶   ▶   𝐴 / 𝑥{𝐵} = { 𝐴 / 𝑥𝐵}   )
5:4: (   𝐴𝐶   ▶   (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})   )
6:3,5: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹 “ {𝐵 }) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})   )
7:1: (   𝐴𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ 𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦})   )
8:6,2: (   𝐴𝐶   ▶   (𝐴 / 𝑥(𝐹 “ { 𝐵}) = 𝐴 / 𝑥{𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
9:7,8: (   𝐴𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})    )
10:9: (   𝐴𝐶   ▶   𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
11:10: (   𝐴𝐶   ▶   {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
12:1: (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   )
13:11,12: (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦 }}   )
14:13: (   𝐴𝐶   ▶    𝐴 / 𝑥{𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 {𝐴 / 𝑥𝐵}) = {𝑦}}   )
15:1: (   𝐴𝐶   ▶   𝐴 / 𝑥 {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
16:14,15: (   𝐴𝐶   ▶   𝐴 / 𝑥 {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
17:: (𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
18:17: 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵 }) = {𝑦}}
19:1,18: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
20:16,19: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
21:: (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}
22:20,21: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
qed:22: (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbfv12gALTVD (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

Proof of Theorem csbfv12gALTVD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 idn1 39308 . . . . . . . . . . 11 (   𝐴𝐶   ▶   𝐴𝐶   )
2 sbceqg 4128 . . . . . . . . . . 11 (𝐴𝐶 → ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ 𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦}))
31, 2e1a 39370 . . . . . . . . . 10 (   𝐴𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ 𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦})   )
4 csbima12 5622 . . . . . . . . . . . . . 14 𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵})
54a1i 11 . . . . . . . . . . . . 13 (𝐴𝐶𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}))
61, 5e1a 39370 . . . . . . . . . . . 12 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵})   )
7 csbsng 4380 . . . . . . . . . . . . . 14 (𝐴𝐶𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
81, 7e1a 39370 . . . . . . . . . . . . 13 (   𝐴𝐶   ▶   𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵}   )
9 imaeq2 5601 . . . . . . . . . . . . 13 (𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵} → (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}))
108, 9e1a 39370 . . . . . . . . . . . 12 (   𝐴𝐶   ▶   (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})   )
11 eqeq1 2775 . . . . . . . . . . . . 13 (𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) → (𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) ↔ (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})))
1211biimprd 238 . . . . . . . . . . . 12 (𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) → ((𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) → 𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})))
136, 10, 12e11 39431 . . . . . . . . . . 11 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})   )
14 csbconstg 3695 . . . . . . . . . . . 12 (𝐴𝐶𝐴 / 𝑥{𝑦} = {𝑦})
151, 14e1a 39370 . . . . . . . . . . 11 (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦} = {𝑦}   )
16 eqeq12 2784 . . . . . . . . . . . 12 ((𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) ∧ 𝐴 / 𝑥{𝑦} = {𝑦}) → (𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}))
1716ex 397 . . . . . . . . . . 11 (𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) → (𝐴 / 𝑥{𝑦} = {𝑦} → (𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})))
1813, 15, 17e11 39431 . . . . . . . . . 10 (   𝐴𝐶   ▶   (𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
19 bibi1 340 . . . . . . . . . . 11 (([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ 𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦}) → (([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}) ↔ (𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})))
2019biimprd 238 . . . . . . . . . 10 (([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ 𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦}) → ((𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}) → ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})))
213, 18, 20e11 39431 . . . . . . . . 9 (   𝐴𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
2221gen11 39359 . . . . . . . 8 (   𝐴𝐶   ▶   𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
23 abbi 2886 . . . . . . . . 9 (∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}) ↔ {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}})
2423biimpi 206 . . . . . . . 8 (∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}) → {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}})
2522, 24e1a 39370 . . . . . . 7 (   𝐴𝐶   ▶   {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
26 csbab 4152 . . . . . . . . 9 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}
2726a1i 11 . . . . . . . 8 (𝐴𝐶𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}})
281, 27e1a 39370 . . . . . . 7 (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   )
29 eqeq2 2782 . . . . . . . 8 ({𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → (𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} ↔ 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}))
3029biimpd 219 . . . . . . 7 ({𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → (𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} → 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}))
3125, 28, 30e11 39431 . . . . . 6 (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
32 unieq 4582 . . . . . 6 (𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}})
3331, 32e1a 39370 . . . . 5 (   𝐴𝐶   ▶    𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
34 csbuni 4602 . . . . . . 7 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
3534a1i 11 . . . . . 6 (𝐴𝐶𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}})
361, 35e1a 39370 . . . . 5 (   𝐴𝐶   ▶   𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
37 eqeq2 2782 . . . . . 6 ( 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → (𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} ↔ 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}))
3837biimpd 219 . . . . 5 ( 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → (𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} → 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}))
3933, 36, 38e11 39431 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
40 dffv4 6327 . . . . . 6 (𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
4140ax-gen 1870 . . . . 5 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
42 csbeq2 3686 . . . . . 6 (∀𝑥(𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} → 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}})
4342a1i 11 . . . . 5 (𝐴𝐶 → (∀𝑥(𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} → 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}))
441, 41, 43e10 39437 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
45 eqeq2 2782 . . . . 5 (𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → (𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} ↔ 𝐴 / 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}))
4645biimpd 219 . . . 4 (𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → (𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} → 𝐴 / 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}))
4739, 44, 46e11 39431 . . 3 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
48 dffv4 6327 . . 3 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}
49 eqeq2 2782 . . . 4 ((𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → (𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}))
5049biimprcd 240 . . 3 (𝐴 / 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → ((𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}} → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
5147, 48, 50e10 39437 . 2 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
5251in1 39305 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629   = wceq 1631  wcel 2145  {cab 2757  [wsbc 3587  csb 3682  {csn 4316   cuni 4574  cima 5252  cfv 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fv 6037  df-vd1 39304
This theorem is referenced by: (None)
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