Theorem vtoclf

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

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

Proof

Step	Hyp	Ref	Expression
1		vtoclf.1	⊢ Ⅎ 𝑥 𝜓
2		vtoclf.2	⊢ 𝐴 ∈ V
3		vtoclf.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4		vtoclf.4	⊢ 𝜑
5	2	isseti	⊢ ∃ 𝑥 𝑥 = 𝐴
6	3	biimpd	⊢ ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) )
7	5 6	eximii	⊢ ∃ 𝑥 ( 𝜑 → 𝜓 )
8	1 7	19.36i	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )
9	8 4	mpg	⊢ 𝜓