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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-19.2reqv | Structured version Visualization version GIF version |
Description: Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1979 is provable from Tarski's FOL and ax13v 2371 only. Note that in conjunction with 19.2 1979 in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) |
Ref | Expression |
---|---|
wl-19.2reqv | ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax13lem2 2374 | . 2 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | |
2 | ax13lem1 2372 | . 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
3 | 1, 2 | syld 47 | 1 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1781 |
This theorem is referenced by: (None) |
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