Theorem pm13.14 41700

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 3705	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	biimprcd 253	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → (𝑥 = 𝐴 → 𝜑))
3	2	necon3bd 2954	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → (¬ 𝜑 → 𝑥 ≠ 𝐴))
4	3	imp 410	1 ⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543   ≠ wne 2940  [wsbc 3694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-sbc 3695

This theorem is referenced by: (None)