Theorem vtoclbg

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994)

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

Proof

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