Theorem sbceq1dd 2961

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

Hypotheses

Ref	Expression
sbceq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
sbceq1dd.2	⊢ (𝜑 → [𝐴 / 𝑥]𝜓)

Assertion

Ref	Expression
sbceq1dd	⊢ (𝜑 → [𝐵 / 𝑥]𝜓)

Proof of Theorem sbceq1dd

Step	Hyp	Ref	Expression
1		sbceq1dd.2	. 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
2		sbceq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	sbceq1d 2960	. 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
4	1, 3	mpbid 146	1 ⊢ (𝜑 → [𝐵 / 𝑥]𝜓)

Syntax hints:   → wi 4   = wceq 1348  [wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-sbc 2956

This theorem is referenced by:  prmind2  12074