Theorem pm13.14 39130

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 3587	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	biimprcd 240	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → (𝑥 = 𝐴 → 𝜑))
3	2	necon3bd 2946	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → (¬ 𝜑 → 𝑥 ≠ 𝐴))
4	3	imp 444	1 ⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1632   ≠ wne 2932  [wsbc 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ne 2933  df-sbc 3577

This theorem is referenced by: (None)