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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-19.8eqv | Structured version Visualization version GIF version |
Description: Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2170 is provable from Tarski's FOL and ax13v 2382 only. Note that this reverts the implication in ax13lem2 2385, so in fact (¬ 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) |
Ref | Expression |
---|---|
wl-19.8eqv | ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax13lem1 2383 | . 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
2 | 19.2 1972 | . 2 ⊢ (∀𝑥 𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦) | |
3 | 1, 2 | syl6 35 | 1 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 |
This theorem is referenced by: (None) |
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