Theorem sbceq1dd 3797

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 3796	. 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → [𝐵 / 𝑥]𝜓)

Syntax hints:   → wi 4   = wceq 1537  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814  df-sbc 3792

This theorem is referenced by:  prmind2  16719  sdclem2  37729  sbceq1ddi  38110