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Mirrors > Home > ILE Home > Th. List > rexlimiva | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.) |
Ref | Expression |
---|---|
rexlimiva.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) |
Ref | Expression |
---|---|
rexlimiva | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimiva.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) | |
2 | 1 | ex 114 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
3 | 2 | rexlimiv 2575 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 ∃wrex 2443 |
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 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-ral 2447 df-rex 2448 |
This theorem is referenced by: unon 4482 reg2exmidlema 4505 ssfilem 6832 diffitest 6844 fival 6926 elfi2 6928 fi0 6931 djuss 7026 updjud 7038 enumct 7071 finnum 7130 dmaddpqlem 7309 nqpi 7310 nq0nn 7374 recexprlemm 7556 rexanuz 10916 r19.2uz 10921 maxleast 11141 fsum2dlemstep 11361 fisumcom2 11365 fprod2dlemstep 11549 fprodcom2fi 11553 0dvds 11737 even2n 11796 m1expe 11821 m1exp1 11823 modprm0 12165 epttop 12637 neipsm 12701 tgioo 13093 sin0pilem2 13250 pilem3 13251 bj-nn0suc 13687 bj-nn0sucALT 13701 trirec0xor 13765 |
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