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Mirrors > Home > ILE Home > Th. List > rexlimivv | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
rexlimivv.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
rexlimivv | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimivv.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)) | |
2 | 1 | rexlimdva 2594 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → 𝜓)) |
3 | 2 | rexlimiv 2588 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 ∃wrex 2456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-ral 2460 df-rex 2461 |
This theorem is referenced by: opelxp 4658 f1o2ndf1 6231 xpdom2 6833 distrlem5prl 7587 distrlem5pru 7588 mulrid 7956 cnegex 8137 recexap 8612 creur 8918 creui 8919 cju 8920 elz2 9326 qre 9627 qaddcl 9637 qnegcl 9638 qmulcl 9639 qreccl 9644 elpqb 9651 replim 10870 prodmodc 11588 odd2np1 11880 opoe 11902 omoe 11903 opeo 11904 omeo 11905 qredeu 12099 pythagtriplem1 12267 pcz 12333 4sqlem1 12388 4sqlem2 12389 4sqlem4 12392 mul4sq 12394 txuni2 13841 blssioo 14130 tgioo 14131 2sqlem2 14547 mul2sq 14548 2sqlem7 14553 |
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