Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > rexlimivv | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
rexlimivv.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
rexlimivv | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimivv.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓)) | |
2 | 1 | rexlimdva 2526 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → 𝜓)) |
3 | 2 | rexlimiv 2520 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ 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: opelxp 4539 f1o2ndf1 6093 xpdom2 6693 distrlem5prl 7362 distrlem5pru 7363 mulid1 7731 cnegex 7908 recexap 8382 creur 8685 creui 8686 cju 8687 elz2 9090 qre 9385 qaddcl 9395 qnegcl 9396 qmulcl 9397 qreccl 9402 replim 10599 odd2np1 11497 opoe 11519 omoe 11520 opeo 11521 omeo 11522 qredeu 11705 txuni2 12352 blssioo 12641 tgioo 12642 |
Copyright terms: Public domain | W3C validator |