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Mirrors > Home > NFE Home > Th. List > reueq1 | GIF version |
Description: Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
Ref | Expression |
---|---|
reueq1 | ⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2489 | . 2 ⊢ ℲxA | |
2 | nfcv 2489 | . 2 ⊢ ℲxB | |
3 | 1, 2 | reueq1f 2805 | 1 ⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∃!wreu 2616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-cleq 2346 df-clel 2349 df-nfc 2478 df-reu 2621 |
This theorem is referenced by: reueqd 2817 |
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