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Theorem pm13.194 37635
 Description: Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.194 ((𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦))

Proof of Theorem pm13.194
StepHypRef Expression
1 pm13.13a 37630 . . . 4 ((𝜑𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
2 sbsbc 3406 . . . 4 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
31, 2sylibr 223 . . 3 ((𝜑𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
4 simpl 472 . . 3 ((𝜑𝑥 = 𝑦) → 𝜑)
5 simpr 476 . . 3 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
63, 4, 53jca 1235 . 2 ((𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦))
7 3simpc 1053 . 2 (([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
86, 7impbii 198 1 ((𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  [wsb 1867  [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-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403 This theorem is referenced by: (None)
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