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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 2663 | . 2 ⊢ (∃!𝑥𝜓 → ∃*𝑥𝜓) | |
2 | moim 2626 | . 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 1535 ∃*wmo 2620 ∃!weu 2653 |
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 1911 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-mo 2622 df-eu 2654 |
This theorem is referenced by: euim 2701 euimOLD 2702 2eumo 2727 moeq3 3703 reuss2 4283 |
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