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Theorem sbid2h 1778
 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 1739 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
31sbh 1707 . 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 1288  [wsb 1693 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473 This theorem depends on definitions:  df-bi 116  df-sb 1694 This theorem is referenced by:  sbid2  1779  sb5rf  1781  sb6rf  1782  sbid2v  1921
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