Theorem pm13.14 42027

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 3727	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	biimprcd 249	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → (𝑥 = 𝐴 → 𝜑))
3	2	necon3bd 2957	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → (¬ 𝜑 → 𝑥 ≠ 𝐴))
4	3	imp 407	1 ⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ≠ wne 2943  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-sbc 3717

This theorem is referenced by: (None)