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Mirrors > Home > NFE Home > Th. List > r19.41v | GIF version |
Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.) |
Ref | Expression |
---|---|
r19.41v | ⊢ (∃x ∈ A (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
2 | 1 | r19.41 2763 | 1 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃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-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-nf 1545 df-rex 2620 |
This theorem is referenced by: r19.42v 2765 3reeanv 2779 reuind 3039 iuncom4 3976 dfiun2g 3999 iunxiun 4048 elpw12 4145 pw1in 4164 imacok 4282 unipw1 4325 dfaddc2 4381 addcass 4415 ltfinex 4464 ncfinlowerlem1 4482 dfevenfin2 4512 dfoddfin2 4513 nnpweqlem1 4522 vfinspss 4551 vfinncsp 4554 phialllem1 4616 setconslem6 4736 xpiundi 4817 xpiundir 4818 elimapw1 4944 elimapw12 4945 elimapw13 4946 imaco 5086 coiun 5090 abrexco 5463 imaiun 5464 isomin 5496 isoini 5497 xpassen 6057 enpw1pw 6075 |
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