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Theorem pm13.196a 37637
 Description: Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.196a 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦𝑥))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm13.196a
StepHypRef Expression
1 sbelx 2446 . 2 𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥] ¬ 𝜑))
2 sb56 2136 . 2 (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑))
3 sbn 2379 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
43imbi2i 325 . . . 4 ((𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦 = 𝑥 → ¬ [𝑦 / 𝑥]𝜑))
5 con2b 348 . . . 4 ((𝑦 = 𝑥 → ¬ [𝑦 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → ¬ 𝑦 = 𝑥))
6 df-ne 2782 . . . . . 6 (𝑦𝑥 ↔ ¬ 𝑦 = 𝑥)
76bicomi 213 . . . . 5 𝑦 = 𝑥𝑦𝑥)
87imbi2i 325 . . . 4 (([𝑦 / 𝑥]𝜑 → ¬ 𝑦 = 𝑥) ↔ ([𝑦 / 𝑥]𝜑𝑦𝑥))
94, 5, 83bitri 285 . . 3 ((𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ([𝑦 / 𝑥]𝜑𝑦𝑥))
109albii 1737 . 2 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦𝑥))
111, 2, 103bitri 285 1 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  [wsb 1867   ≠ wne 2780 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 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-ne 2782 This theorem is referenced by: (None)
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