Theorem csbsngVD 46066

Description: Virtual deduction proof of csbsng 4669. 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. csbsng 4669 is csbsngVD 46066 without virtual deductions and was automatically derived from csbsngVD 46066.

1::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   )
4:3:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
5:2,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
6:5:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
7:6:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
8:1:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
9:7,8:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
10::	⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
11:10:	⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
12:1,11:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = ⦋ 𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
13:9,12:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { 𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
14::	⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
15:13,14:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { ⦋𝐴 / 𝑥⦌𝐵}   )
qed:15:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋ 𝐴 / 𝑥⦌𝐵})

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbsngVD	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

Proof of Theorem csbsngVD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn1 45748	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2		sbceqg 4373	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
3	1, 2	e1a 45801	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
4		csbconstg 3870	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
5	1, 4	e1a 45801	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   )
6		eqeq1 2767	. . . . . . . . 9 ⊢ (⦋𝐴 / 𝑥⦌𝑦 = 𝑦 → (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
7	5, 6	e1a 45801	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
8		bibi1 354	. . . . . . . . 9 ⊢ (([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵) → (([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) ↔ (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)))
9	8	biimprd 251	. . . . . . . 8 ⊢ (([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵) → ((⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)))
10	3, 7, 9	e11 45862	. . . . . . 7 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
11	10	gen11 45790	. . . . . 6 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   )
12		abbib 2832	. . . . . . 7 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} ↔ ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
13	12	biimpri 231	. . . . . 6 ⊢ (∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵) → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
14	11, 13	e1a 45801	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
15		csbab 4401	. . . . . . . 8 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵}
16	15	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵})
17	16	eqcomd 2769	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵})
18	1, 17	e1a 45801	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
19		eqeq1 2767	. . . . . 6 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} → ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} ↔ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
20	19	biimpcd 252	. . . . 5 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → ({𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
21	14, 18, 20	e11 45862	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
22		df-sn 4589	. . . . . 6 ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
23	22	ax-gen 1828	. . . . 5 ⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
24		csbeq2 3856	. . . . . 6 ⊢ (∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵} → ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵})
25	24	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵} → ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}))
26	1, 23, 25	e10 45868	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   )
27		eqeq2 2775	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} ↔ ⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
28	27	biimpd 232	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} → ⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
29	21, 26, 28	e11 45862	. . 3 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   )
30		df-sn 4589	. . 3 ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
31		eqeq2 2775	. . . 4 ⊢ ({⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵} ↔ ⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}))
32	31	biimprcd 253	. . 3 ⊢ (⦋𝐴 / 𝑥⦌{𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → ({⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}))
33	29, 30, 32	e10 45868	. 2 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}   )
34	33	in1 45745	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  [wsbc 3742  ⦋csb 3851  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456  df-sbc 3743  df-csb 3852  df-dif 4050  df-nul 4282  df-sn 4589  df-vd1 45744

This theorem is used by: (None)