Theorem vtocl 2856

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)

Hypotheses

Ref	Expression
vtocl.1	⊢ 𝐴 ∈ V
vtocl.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl.3	⊢ 𝜑

Assertion

Ref	Expression
vtocl	⊢ 𝜓

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtocl

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜓
2		vtocl.1	. 2 ⊢ 𝐴 ∈ V
3		vtocl.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		vtocl.3	. 2 ⊢ 𝜑
5	1, 2, 3, 4	vtoclf 2855	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   ∈ wcel 2200  Vcvv 2800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2802

This theorem is referenced by:  vtoclb  2859  zfauscl  4207  bnd2  4261  uniex  4532  ordtriexmid  4617  onsucsssucexmid  4623  regexmid  4631  ordsoexmid  4658  onintexmid  4669  reg3exmid  4676  nnregexmid  4717  acexmidlemv  6011  caovcan  6182  findcard2  7071  findcard2s  7072  inffiexmid  7091  sup3exmid  9127  bj-uniex  16448  bj-omtrans  16487