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Mirrors > Home > ILE Home > Th. List > rexlimdvv | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.) |
Ref | Expression |
---|---|
rexlimdvv.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
rexlimdvv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimdvv.1 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))) | |
2 | 1 | expdimp 259 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝜓 → 𝜒))) |
3 | 2 | rexlimdv 2606 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
4 | 3 | rexlimdva 2607 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 ∃wrex 2469 |
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 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-ral 2473 df-rex 2474 |
This theorem is referenced by: rexlimdvva 2615 f1oiso2 5844 xpdom2 6849 genpcdl 7537 genpcuu 7538 distrlem1prl 7600 distrlem1pru 7601 distrlem5prl 7604 distrlem5pru 7605 recexprlemss1l 7653 recexprlemss1u 7654 qaddcl 9654 qmulcl 9656 summodc 11410 dvdsgcd 12032 gcddiv 12039 pceu 12314 pcqcl 12325 txcnp 14174 blssps 14330 blss 14331 tgqioo 14450 |
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