Theorem sbceq1dd 3748

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 3747	. 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → [𝐵 / 𝑥]𝜓)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-clel 2839  df-sbc 3743

This theorem is used by:  prmind2  16886  sdclem2  38867  sbceq1ddi  39246