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Theorem csbfv12gALTVD 41218
Description: Virtual deduction proof of csbfv12 6706. 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 6706 is csbfv12gALTVD 41218 without virtual deductions and was automatically derived from csbfv12gALTVD 41218.
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 40893 . . . . . . . . . . 11 (   𝐴𝐶   ▶   𝐴𝐶   )
2 sbceqg 4359 . . . . . . . . . . 11 (𝐴𝐶 → ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ 𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦}))
31, 2e1a 40946 . . . . . . . . . 10 (   𝐴𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ 𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦})   )
4 csbima12 5940 . . . . . . . . . . . . . 14 𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵})
54a1i 11 . . . . . . . . . . . . 13 (𝐴𝐶𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}))
61, 5e1a 40946 . . . . . . . . . . . 12 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵})   )
7 csbsng 4636 . . . . . . . . . . . . . 14 (𝐴𝐶𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
81, 7e1a 40946 . . . . . . . . . . . . 13 (   𝐴𝐶   ▶   𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵}   )
9 imaeq2 5918 . . . . . . . . . . . . 13 (𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵} → (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}))
108, 9e1a 40946 . . . . . . . . . . . 12 (   𝐴𝐶   ▶   (𝐴 / 𝑥𝐹𝐴 / 𝑥{𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})   )
11 eqeq1 2823 . . . . . . . . . . . . 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 250 . . . . . . . . . . . 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 41007 . . . . . . . . . . 11 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹 “ {𝐵}) = (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵})   )
14 csbconstg 3900 . . . . . . . . . . . 12 (𝐴𝐶𝐴 / 𝑥{𝑦} = {𝑦})
151, 14e1a 40946 . . . . . . . . . . 11 (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦} = {𝑦}   )
16 eqeq12 2833 . . . . . . . . . . . 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 415 . . . . . . . . . . 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 41007 . . . . . . . . . 10 (   𝐴𝐶   ▶   (𝐴 / 𝑥(𝐹 “ {𝐵}) = 𝐴 / 𝑥{𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
19 bibi1 354 . . . . . . . . . . 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 250 . . . . . . . . . 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 41007 . . . . . . . . 9 (   𝐴𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦})   )
2221gen11 40935 . . . . . . . 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 218 . . . . . . . 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 40946 . . . . . . 7 (   𝐴𝐶   ▶   {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
26 csbab 4387 . . . . . . . . 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 40946 . . . . . . 7 (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦[𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   )
29 eqeq2 2831 . . . . . . . 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 231 . . . . . . 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 41007 . . . . . 6 (   𝐴𝐶   ▶   𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
32 unieq 4838 . . . . . 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 40946 . . . . 5 (   𝐴𝐶   ▶    𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
34 csbuni 4858 . . . . . . 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 40946 . . . . 5 (   𝐴𝐶   ▶   𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = 𝐴 / 𝑥{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
37 eqeq2 2831 . . . . . 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 231 . . . . 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 41007 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
40 dffv4 6660 . . . . . 6 (𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
4140ax-gen 1789 . . . . 5 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
42 csbeq2 3886 . . . . . 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 41013 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥 {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
45 eqeq2 2831 . . . . 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 231 . . . 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 41007 . . 3 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}   )
48 dffv4 6660 . . 3 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = {𝑦 ∣ (𝐴 / 𝑥𝐹 “ {𝐴 / 𝑥𝐵}) = {𝑦}}
49 eqeq2 2831 . . . 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 252 . . 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 41013 . 2 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
5251in1 40890 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1528   = wceq 1530  wcel 2107  {cab 2797  [wsbc 3770  csb 3881  {csn 4559   cuni 4830  cima 5551  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-vd1 40889
This theorem is referenced by: (None)
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