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Theorem pm13.195 44432
Description: Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3816. (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 3816 . 2 ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴𝜑))
21bicomi 224 1 (∃𝑦(𝑦 = 𝐴𝜑) ↔ [𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789
This theorem is referenced by: (None)
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