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Mirrors > Home > NFE Home > Th. List > rexcom | GIF version |
Description: Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
rexcom | ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2489 | . 2 ⊢ ℲyA | |
2 | nfcv 2489 | . 2 ⊢ ℲxB | |
3 | 1, 2 | rexcomf 2770 | 1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wrex 2615 |
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-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 |
This theorem is referenced by: rexcom13 2773 rexcom4 2878 iuncom 3975 addccom 4406 ltfinex 4464 nnpw1ex 4484 evenodddisjlem1 4515 phialllem1 4616 xpiundi 4817 addcfnex 5824 crossex 5850 mucex 6133 ncspw1eu 6159 addlec 6208 nc0le1 6216 dflec3 6221 |
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