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Theorem sbid2 2530
Description: An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2382. Check out sbid2vw 2259 for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
sbid2.1 𝑥𝜑
Assertion
Ref Expression
sbid2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Proof of Theorem sbid2
StepHypRef Expression
1 sbco 2529 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
2 sbid2.1 . . 3 𝑥𝜑
32sbf 2270 . 2 ([𝑦 / 𝑥]𝜑𝜑)
41, 3bitri 278 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wnf 1785  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by:  sbid2v  2531  sbtrt  2537
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