Theorem vtoclgf 3250

Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
vtoclgf.1	⊢ Ⅎ𝑥𝐴
vtoclgf.2	⊢ Ⅎ𝑥𝜓
vtoclgf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtoclgf.4	⊢ 𝜑

Assertion

Ref	Expression
vtoclgf	⊢ (𝐴 ∈ 𝑉 → 𝜓)

Proof of Theorem vtoclgf

Step	Hyp	Ref	Expression
1		elex 3198	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtoclgf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 3194	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		vtoclgf.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5		vtoclgf.4	. . . . 5 ⊢ 𝜑
6		vtoclgf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	mpbii 223	. . . 4 ⊢ (𝑥 = 𝐴 → 𝜓)
8	4, 7	exlimi 2084	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
9	3, 8	sylbi 207	. 2 ⊢ (𝐴 ∈ V → 𝜓)
10	1, 9	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480  ∃wex 1701  Ⅎwnf 1705   ∈ wcel 1987  Ⅎwnfc 2748  Vcvv 3186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188

This theorem is referenced by:  vtocl2gf  3254  vtocl3gf  3255  vtoclgaf  3257  elabgf  3331  fprodsplit1f  14646  ssiun2sf  29220  subtr  31947  subtr2  31948  supxrgere  39010  supxrgelem  39014  supxrge  39015  fsumsplit1  39205  fmuldfeqlem1  39215  fprodcnlem  39232  climsuse  39241  dvnmptdivc  39456  dvmptfprodlem  39462  stoweidlem59  39580  fourierdlem31  39659  sge0f1o  39903  sge0fodjrnlem  39937  salpreimagelt  40222  salpreimalegt  40224