Theorem vtoclALT 3544

Description: Alternate proof of vtocl 3536. Shorter but requires more axioms. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
vtoclALT	⊢ 𝜓

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclALT

Step	Hyp	Ref	Expression
1		nfv 1909	. 2 ⊢ Ⅎ𝑥𝜓
2		vtoclALT.1	. 2 ⊢ 𝐴 ∈ V
3		vtoclALT.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		vtoclALT.3	. 2 ⊢ 𝜑
5	1, 2, 3, 4	vtoclf 3542	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778  df-clel 2802

This theorem is referenced by: (None)