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Theorem sbid2 2503
Description: An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2367. Check out sbid2vw 2246 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 2502 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
2 sbid2.1 . . 3 𝑥𝜑
32sbf 2258 . 2 ([𝑦 / 𝑥]𝜑𝜑)
41, 3bitri 275 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wnf 1778  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-12 2167  ax-13 2367
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779  df-sb 2061
This theorem is referenced by:  sbid2v  2504  sbtrt  2510
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