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Theorem pm13.192 38128
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 1736 . . . . . 6 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝑥 = 𝐴))
3 eqeq1 2625 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
43equsalvw 1928 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
52, 4sylib 208 . . . . 5 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) → 𝑦 = 𝐴)
6 eqeq2 2632 . . . . . . 7 (𝐴 = 𝑦 → (𝑥 = 𝐴𝑥 = 𝑦))
76eqcoms 2629 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝑦))
87alrimiv 1852 . . . . 5 (𝑦 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥 = 𝑦))
95, 8impbii 199 . . . 4 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ↔ 𝑦 = 𝐴)
109anbi1i 730 . . 3 ((∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ (𝑦 = 𝐴𝜑))
1110exbii 1771 . 2 (∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴𝜑))
12 sbc5 3446 . 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 384  wal 1478   = wceq 1480  wex 1701  [wsbc 3421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3191  df-sbc 3422
This theorem is referenced by: (None)
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