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Metamath Proof Explorer

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 ) )