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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 1493 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1493 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | rexlimdv.1 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
4 | 1, 2, 3 | rexlimd 2523 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ∃wrex 2394 |
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 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-i5r 1500 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-ral 2398 df-rex 2399 |
This theorem is referenced by: rexlimdva 2526 rexlimdv3a 2528 rexlimdva2 2529 rexlimdvw 2530 rexlimdvv 2533 ssorduni 4373 funcnvuni 5162 dffo3 5535 smoiun 6166 tfrlem9 6184 ordiso2 6888 axprecex 7656 recexap 8382 zdiv 9107 btwnz 9138 lbzbi 9376 neibl 12587 metcnp3 12607 |
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