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Theorem pm13.195 37439
Description: Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3426. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
Assertion
Ref Expression
pm13.195 (∃𝑦(𝑦 = 𝐴𝜑) ↔ [𝐴 / 𝑦]𝜑)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem pm13.195
StepHypRef Expression
1 sbc5 3426 . 2 ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴𝜑))
21bicomi 212 1 (∃𝑦(𝑦 = 𝐴𝜑) ↔ [𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wex 1694  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-v 3174  df-sbc 3402
This theorem is referenced by: (None)
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