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Theorem sbequ12a 1698
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
StepHypRef Expression
1 sbequ12 1696 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
2 sbequ12 1696 . . 3 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
32equcoms 1636 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
41, 3bitr3d 188 1 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464
This theorem depends on definitions:  df-bi 115  df-sb 1688
This theorem is referenced by: (None)
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