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Metamath Proof Explorer

Theorem sbceq1dd

Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017) (Revised by NM, 30-Jun-2018)

Ref Expression
Hypotheses sbceq1d.1 ⊢ φ → A = B
sbceq1dd.2 ⊢ φ → [˙A / x]˙ ψ
Assertion sbceq1dd ⊢ φ → [˙B / x]˙ ψ

Proof

Step Hyp Ref Expression
1 sbceq1d.1 ⊢ φ → A = B
2 sbceq1dd.2 ⊢ φ → [˙A / x]˙ ψ
3 1 sbceq1d ⊢ φ → [˙A / x]˙ ψ ↔ [˙B / x]˙ ψ
4 2 3 mpbid ⊢ φ → [˙B / x]˙ ψ