Theorem vtoclgf 3521

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 3457	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtoclgf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 3453	. . 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 2220	. . 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 1541  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2111  Ⅎwnfc 2879  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-v 3438

This theorem is referenced by:  vtocl2gf  3523  vtocl3gf  3524  vtoclgaf  3527  elabgf  3625  fsumsplit1  15652  ssiun2sf  32539  subtr  36358  subtr2  36359  supxrgere  45431  supxrgelem  45435  supxrge  45436  fmuldfeqlem1  45681  climsuse  45707  dvnmptdivc  46035  dvmptfprodlem  46041  stoweidlem59  46156  fourierdlem31  46235  sge0fodjrnlem  46513