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Mirrors > Home > ILE Home > Th. List > rexlimi | GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
rexlimi.1 | ⊢ Ⅎ𝑥𝜓 |
rexlimi.2 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
rexlimi | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimi.2 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
2 | 1 | rgen 2517 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) |
3 | rexlimi.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | r19.23 2572 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
5 | 2, 4 | mpbi 144 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1447 ∈ wcel 2135 ∀wral 2442 ∃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-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: rexlimiv 2575 r19.29af2 2604 triun 4087 reusv1 4430 reusv3 4432 onintrab2im 4489 fun11iun 5447 fisumcom2 11365 fprodcom2fi 11553 |
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