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Theorem pm13.192 40965
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 223 . . . . . . 7 ((𝑥 = 𝐴𝑥 = 𝑦) → (𝑥 = 𝑦𝑥 = 𝐴))
21alimi 1813 . . . . . 6 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝑥 = 𝐴))
3 eqeq1 2828 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
43equsalvw 2011 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
52, 4sylib 221 . . . . 5 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) → 𝑦 = 𝐴)
6 eqeq2 2836 . . . . . . 7 (𝐴 = 𝑦 → (𝑥 = 𝐴𝑥 = 𝑦))
76eqcoms 2832 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝑦))
87alrimiv 1929 . . . . 5 (𝑦 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥 = 𝑦))
95, 8impbii 212 . . . 4 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ↔ 𝑦 = 𝐴)
109anbi1i 626 . . 3 ((∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ (𝑦 = 𝐴𝜑))
1110exbii 1849 . 2 (∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴𝜑))
12 sbc5 3785 . 2 ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴𝜑))
1311, 12bitr4i 281 1 (∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  [wsbc 3757
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-sbc 3758
This theorem is referenced by: (None)
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