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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm13.14 | Structured version Visualization version GIF version |
Description: Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
pm13.14 | ⊢ (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥 ≠ 𝐴) |
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) |
Colors of variables: wff setvar class |
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) |
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