Theorem sbceq1d 3051

Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypothesis

Ref	Expression
sbceq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
sbceq1d	⊢ (φ → ([̣A / x]̣φ ↔ [̣B / x]̣φ))

Proof of Theorem sbceq1d

Step	Hyp	Ref	Expression
1		sbceq1d.1	. 2 ⊢ (φ → A = B)
2		dfsbcq 3048	. 2 ⊢ (A = B → ([̣A / x]̣φ ↔ [̣B / x]̣φ))
3	1, 2	syl 15	1 ⊢ (φ → ([̣A / x]̣φ ↔ [̣B / x]̣φ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [̣wsbc 3046

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-cleq 2346  df-clel 2349  df-sbc 3047

This theorem is referenced by:  sbceq1dd  3052