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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 2603 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
4 | 3 | rexlimdva 2604 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2158 ∃wrex 2466 |
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 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-4 1520 ax-17 1536 ax-ial 1544 ax-i5r 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-ral 2470 df-rex 2471 |
This theorem is referenced by: rexlimdvva 2612 f1oiso2 5841 xpdom2 6844 genpcdl 7531 genpcuu 7532 distrlem1prl 7594 distrlem1pru 7595 distrlem5prl 7598 distrlem5pru 7599 recexprlemss1l 7647 recexprlemss1u 7648 qaddcl 9648 qmulcl 9650 summodc 11404 dvdsgcd 12026 gcddiv 12033 pceu 12308 pcqcl 12319 txcnp 14011 blssps 14167 blss 14168 tgqioo 14287 |
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