Theorem vtoclgf 3529

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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vtoclgf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	issetf 3467	. . 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 236	. . . 4 ⊢ (𝑥 = 𝐴 → 𝜓)
8	4, 7	exlimi 2253	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
9	3, 8	sylbi 220	. 2 ⊢ (𝐴 ∈ V → 𝜓)
10	1, 9	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2145  Ⅎwnfc 2907  Vcvv 3450

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452

This theorem is used by:  vtocl2gf  3531  vtocl3gf  3532  vtoclgaf  3535  elabgf  3628  fsumsplit1  15856  ssiun2sf  33065  subtr  37000  subtr2  37001  supxrgere  46228  supxrgelem  46232  supxrge  46233  fmuldfeqlem1  46477  climsuse  46503  dvnmptdivc  46831  dvmptfprodlem  46837  stoweidlem59  46952  fourierdlem31  47031  sge0fodjrnlem  47309