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Theorem pm13.192 39138
Description: Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
Assertion
Ref Expression
pm13.192 (∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem pm13.192
StepHypRef Expression
1 biimpr 210 . . . . . . 7 ((𝑥 = 𝐴𝑥 = 𝑦) → (𝑥 = 𝑦𝑥 = 𝐴))
21alimi 1887 . . . . . 6 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝑥 = 𝐴))
3 eqeq1 2775 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
43equsalvw 2089 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
52, 4sylib 208 . . . . 5 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) → 𝑦 = 𝐴)
6 eqeq2 2782 . . . . . . 7 (𝐴 = 𝑦 → (𝑥 = 𝐴𝑥 = 𝑦))
76eqcoms 2779 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝑦))
87alrimiv 2007 . . . . 5 (𝑦 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥 = 𝑦))
95, 8impbii 199 . . . 4 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ↔ 𝑦 = 𝐴)
109anbi1i 604 . . 3 ((∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ (𝑦 = 𝐴𝜑))
1110exbii 1924 . 2 (∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴𝜑))
12 sbc5 3613 . 2 ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴𝜑))
1311, 12bitr4i 267 1 (∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wex 1852  [wsbc 3588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3589
This theorem is referenced by: (None)
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