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Mirrors > Home > MPE Home > Th. List > euimmo | Structured version Visualization version GIF version |
Description: Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995.) |
Ref | Expression |
---|---|
euimmo | ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2570 | . 2 ⊢ (∃!𝑥𝜓 → ∃*𝑥𝜓) | |
2 | moim 2536 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑)) | |
3 | 1, 2 | syl5 34 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∃*wmo 2530 ∃!weu 2560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-mo 2532 df-eu 2561 |
This theorem is referenced by: euim 2611 2eumo 2636 moeq3 3709 reuss2 4316 |
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