Theorem vtoclb

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

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

Proof

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