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Theorem pm13.194 37434
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 37429 . . . 4 ((𝜑𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
2 sbsbc 3401 . . . 4 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
31, 2sylibr 222 . . 3 ((𝜑𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑)
4 simpl 471 . . 3 ((𝜑𝑥 = 𝑦) → 𝜑)
5 simpr 475 . . 3 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
63, 4, 53jca 1234 . 2 ((𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦))
7 3simpc 1052 . 2 (([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
86, 7impbii 197 1 ((𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030  [wsb 1865  [wsbc 3397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-ex 1695  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-sbc 3398
This theorem is referenced by: (None)
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