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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 108 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
3 | 2 | rexlimiv 2427 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-ral 2311 df-rex 2312 |
This theorem is referenced by: unon 4237 reg2exmidlema 4259 ssfiexmid 6336 diffitest 6344 finnum 6363 dmaddpqlem 6475 nqpi 6476 nq0nn 6540 recexprlemm 6722 rexanuz 9587 r19.2uz 9591 bj-nn0suc 10089 bj-nn0sucALT 10103 |
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