Theorem sbceq2a 3062

Description: Equality theorem for class substitution. Class version of sbequ12r 1825. (Contributed by NM, 4-Jan-2017.)

Assertion

Ref	Expression
sbceq2a	⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Proof of Theorem sbceq2a

Step	Hyp	Ref	Expression
1		sbceq1a 3061	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	eqcoms 2241	. 2 ⊢ (𝐴 = 𝑥 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	bicomd 141	1 ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  [wsbc 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052

This theorem is referenced by:  uchoice  6365