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Theorem pm13.194 39139
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 39134 . . . 4 ((𝜑𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
2 sbsbc 3591 . . . 4 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
31, 2sylibr 224 . . 3 ((𝜑𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
4 simpl 468 . . 3 ((𝜑𝑥 = 𝑦) → 𝜑)
5 simpr 471 . . 3 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
63, 4, 53jca 1122 . 2 ((𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦))
7 3simpc 1146 . 2 (([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
86, 7impbii 199 1 ((𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071  [wsb 2049  [wsbc 3587
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-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-3an 1073  df-ex 1853  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-sbc 3588
This theorem is referenced by: (None)
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