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Theorem sbid2h 1859
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 1820 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
31sbh 1786 . 2 ([𝑦 / 𝑥]𝜑𝜑)
42, 3bitri 184 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1361  [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544
This theorem depends on definitions:  df-bi 117  df-sb 1773
This theorem is referenced by:  sbid2  1860  sb5rf  1862  sb6rf  1863  sbid2v  2006
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