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Theorem pm13.192 37633
 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 209 . . . . . . 7 ((𝑥 = 𝐴𝑥 = 𝑦) → (𝑥 = 𝑦𝑥 = 𝐴))
21alimi 1730 . . . . . 6 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝑥 = 𝐴))
3 eqeq1 2614 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
43equsalvw 1918 . . . . . 6 (∀𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
52, 4sylib 207 . . . . 5 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) → 𝑦 = 𝐴)
6 eqeq2 2621 . . . . . . 7 (𝐴 = 𝑦 → (𝑥 = 𝐴𝑥 = 𝑦))
76eqcoms 2618 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝐴𝑥 = 𝑦))
87alrimiv 1842 . . . . 5 (𝑦 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥 = 𝑦))
95, 8impbii 198 . . . 4 (∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ↔ 𝑦 = 𝐴)
109anbi1i 727 . . 3 ((∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ (𝑦 = 𝐴𝜑))
1110exbii 1764 . 2 (∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴𝜑))
12 sbc5 3427 . 2 ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴𝜑))
1311, 12bitr4i 266 1 (∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by: (None)
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