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Mirrors > Home > MPE Home > Th. List > eu6im | Structured version Visualization version GIF version |
Description: One direction of eu6 2574 needs fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.) |
Ref | Expression |
---|---|
eu6im | ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exsbim 2005 | . . 3 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | |
2 | 1 | anim1i 615 | . 2 ⊢ ((∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) → (∃𝑥𝜑 ∧ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))) |
3 | eu6lem 2573 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))) | |
4 | eu3v 2570 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))) | |
5 | 2, 3, 4 | 3imtr4i 292 | 1 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 ∃wex 1782 ∃!weu 2568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-mo 2540 df-eu 2569 |
This theorem is referenced by: eufsnlem 46168 |
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