Theorem vtoclgf 3553

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 3485	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtoclgf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 3481	. . 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 2884  Vcvv 3464

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 2708

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 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-v 3466

This theorem is referenced by:  vtocl2gf  3556  vtocl3gf  3557  vtoclgaf  3560  elabgf  3658  fsumsplit1  15766  ssiun2sf  32545  subtr  36337  subtr2  36338  supxrgere  45327  supxrgelem  45331  supxrge  45332  fmuldfeqlem1  45578  climsuse  45604  dvnmptdivc  45934  dvmptfprodlem  45940  stoweidlem59  46055  fourierdlem31  46134  sge0fodjrnlem  46412