Theorem pm13.14 40761

Description: Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.14	⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴)

Proof of Theorem pm13.14

Step	Hyp	Ref	Expression
1		sbceq1a 3783	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	biimprcd 252	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → (𝑥 = 𝐴 → 𝜑))
3	2	necon3bd 3030	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → (¬ 𝜑 → 𝑥 ≠ 𝐴))
4	3	imp 409	1 ⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ≠ wne 3016  [wsbc 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-sbc 3773

This theorem is referenced by: (None)