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Theorem sbid2h 1829
Description: An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
sbid2h.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
sbid2h ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Proof of Theorem sbid2h
StepHypRef Expression
1 sbid2h.1 . . 3 (𝜑 → ∀𝑥𝜑)
21sbcof2 1790 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
31sbh 1756 . 2 ([𝑦 / 𝑥]𝜑𝜑)
42, 3bitri 183 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1333  [wsb 1742
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-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-sb 1743
This theorem is referenced by:  sbid2  1830  sb5rf  1832  sb6rf  1833  sbid2v  1976
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