Theorem sbceq2a 3058

Description: Equality theorem for class substitution. Class version of sbequ12r 1920. (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 3057	. . 3 ⊢ (x = A → (φ ↔ [̣A / x]̣φ))
2	1	eqcoms 2356	. 2 ⊢ (A = x → (φ ↔ [̣A / x]̣φ))
3	2	bicomd 192	1 ⊢ (A = x → ([̣A / x]̣φ ↔ φ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3048

This theorem is referenced by: (None)