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Theorem pm13.196a 39141
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 2606 . 2 𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥] ¬ 𝜑))
2 sb56 2271 . 2 (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑))
3 sbn 2538 . . . . 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 2944 . . . . . 6 (𝑦𝑥 ↔ ¬ 𝑦 = 𝑥)
76bicomi 214 . . . . 5 𝑦 = 𝑥𝑦𝑥)
87imbi2i 325 . . . 4 (([𝑦 / 𝑥]𝜑 → ¬ 𝑦 = 𝑥) ↔ ([𝑦 / 𝑥]𝜑𝑦𝑥))
94, 5, 83bitri 286 . . 3 ((𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ([𝑦 / 𝑥]𝜑𝑦𝑥))
109albii 1895 . 2 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦𝑥))
111, 2, 103bitri 286 1 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629  wex 1852  [wsb 2049  wne 2943
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-10 2174  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858  df-sb 2050  df-ne 2944
This theorem is referenced by: (None)
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