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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 1528 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1528 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | rexlimdv.1 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
4 | 1, 2, 3 | rexlimd 2591 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ 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: rexlimdva 2594 rexlimdv3a 2596 rexlimdva2 2597 rexlimdvw 2598 rexlimdvv 2601 ssorduni 4487 funcnvuni 5286 dffo3 5664 smoiun 6302 tfrlem9 6320 ordiso2 7034 axprecex 7879 recexap 8610 zdiv 9341 btwnz 9372 lbzbi 9616 neibl 13994 metcnp3 14014 |
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