Theorem sbceq1dd 3053

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

Hypotheses

Ref	Expression
sbceq1d.1	⊢ (φ → A = B)
sbceq1dd.2	⊢ (φ → [̣A / x]̣φ)

Assertion

Ref	Expression
sbceq1dd	⊢ (φ → [̣B / x]̣φ)

Proof of Theorem sbceq1dd

Step	Hyp	Ref	Expression
1		sbceq1dd.2	. 2 ⊢ (φ → [̣A / x]̣φ)
2		sbceq1d.1	. . 3 ⊢ (φ → A = B)
3	2	sbceq1d 3052	. 2 ⊢ (φ → ([̣A / x]̣φ ↔ [̣B / x]̣φ))
4	1, 3	mpbid 201	1 ⊢ (φ → [̣B / x]̣φ)

Syntax hints:   → wi 4   = 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-cleq 2346  df-clel 2349  df-sbc 3048

This theorem is referenced by: (None)