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Metamath Proof Explorer

Theorem vtoclga

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995) Avoid ax-10 and ax-11 . (Revised by GG, 20-Aug-2023)

	Ref	Expression
Hypotheses	vtoclga.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	vtoclga.2	⊢ ( 𝑥 ∈ 𝐵 → 𝜑 )
Assertion	vtoclga	⊢ ( 𝐴 ∈ 𝐵 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		vtoclga.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		vtoclga.2	⊢ ( 𝑥 ∈ 𝐵 → 𝜑 )
3		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
4	3 1	imbi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ 𝐵 → 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 → 𝜓 ) ) )
5	4 2	vtoclg	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ 𝐵 → 𝜓 ) )
6	5	pm2.43i	⊢ ( 𝐴 ∈ 𝐵 → 𝜓 )