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Mirrors > Home > ILE Home > Th. List > rexlimdv | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
Ref | Expression |
---|---|
rexlimdv.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
rexlimdv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1508 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | rexlimdv.1 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
4 | 1, 2, 3 | rexlimd 2546 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∃wrex 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-ral 2421 df-rex 2422 |
This theorem is referenced by: rexlimdva 2549 rexlimdv3a 2551 rexlimdva2 2552 rexlimdvw 2553 rexlimdvv 2556 ssorduni 4403 funcnvuni 5192 dffo3 5567 smoiun 6198 tfrlem9 6216 ordiso2 6920 axprecex 7688 recexap 8414 zdiv 9139 btwnz 9170 lbzbi 9408 neibl 12660 metcnp3 12680 |
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