Theorem vtoclb 3502

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)

Hypotheses

Ref	Expression
vtoclb.1	⊢ 𝐴 ∈ V
vtoclb.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
vtoclb.3	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
vtoclb.4	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
vtoclb	⊢ (𝜒 ↔ 𝜃)

Distinct variable groups:   𝑥,𝐴   𝜒,𝑥   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem vtoclb

Step	Hyp	Ref	Expression
1		vtoclb.1	. 2 ⊢ 𝐴 ∈ V
2		vtoclb.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
3		vtoclb.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
4	2, 3	bibi12d 346	. 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃)))
5		vtoclb.4	. 2 ⊢ (𝜑 ↔ 𝜓)
6	1, 4, 5	vtocl 3498	1 ⊢ (𝜒 ↔ 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-clel 2816

This theorem is referenced by:  bnj609  32897