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Mirrors > Home > ILE Home > Th. List > rexlimdv3a | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2432. (Contributed by NM, 7-Jun-2015.) |
Ref | Expression |
---|---|
rexlimdv3a.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
rexlimdv3a | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimdv3a.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) | |
2 | 1 | 3exp 1103 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
3 | 2 | rexlimdv 2432 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 885 ∈ 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-3an 887 df-nf 1350 df-ral 2311 df-rex 2312 |
This theorem is referenced by: resqrtcl 9627 |
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