Theorem vtoclgf 3535

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 3468	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtoclgf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 3464	. . 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 233	. . . 4 ⊢ (𝑥 = 𝐴 → 𝜓)
8	4, 7	exlimi 2218	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
9	3, 8	sylbi 217	. 2 ⊢ (𝐴 ∈ V → 𝜓)
10	1, 9	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2876  Vcvv 3447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-v 3449

This theorem is referenced by:  vtocl2gf  3538  vtocl3gf  3539  vtoclgaf  3542  elabgf  3641  fsumsplit1  15711  ssiun2sf  32488  subtr  36302  subtr2  36303  supxrgere  45329  supxrgelem  45333  supxrge  45334  fmuldfeqlem1  45580  climsuse  45606  dvnmptdivc  45936  dvmptfprodlem  45942  stoweidlem59  46057  fourierdlem31  46136  sge0fodjrnlem  46414