Theorem sbequ12a 1787

Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sbequ12a	⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))

Proof of Theorem sbequ12a

Step	Hyp	Ref	Expression
1		sbequ12 1785	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
2		sbequ12 1785	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
3	2	equcoms 1722	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
4	1, 3	bitr3d 190	1 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 105  [wsb 1776

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-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by: (None)