Theorem sbequ12a 1766

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 1764	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
2		sbequ12 1764	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
3	2	equcoms 1701	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
4	1, 3	bitr3d 189	1 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 104  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by: (None)