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Theorem sbid2h 1772
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 1733 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
31sbh 1701 . 2 ([𝑦 / 𝑥]𝜑𝜑)
42, 3bitri 182 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  [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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1688
This theorem is referenced by:  sbid2  1773  sb5rf  1775  sb6rf  1776  sbid2v  1915
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