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Mirrors > Home > MPE Home > Th. List > euim | Structured version Visualization version GIF version |
Description: Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof shortened by Wolf Lammen, 1-Oct-2023.) |
Ref | Expression |
---|---|
euim | ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euimmo 2699 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑)) | |
2 | exmoeub 2664 | . . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑)) | |
3 | 2 | biimpd 231 | . 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑)) |
4 | 1, 3 | sylan9r 511 | 1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1534 ∃wex 1779 ∃*wmo 2619 ∃!weu 2652 |
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 1910 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-mo 2621 df-eu 2653 |
This theorem is referenced by: 2eu1 2734 2eu1OLD 2735 2eu1v 2736 dfatcolem 43461 |
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