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Mirrors > Home > ILE Home > Th. List > rexim | GIF version |
Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
rexim | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2311 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
2 | simpl 102 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴) | |
3 | 2 | a1i 9 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)) |
4 | pm3.31 249 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) | |
5 | 3, 4 | jcad 291 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
6 | 5 | alimi 1344 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
7 | 1, 6 | sylbi 114 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
8 | exim 1490 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))) |
10 | df-rex 2312 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
11 | df-rex 2312 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
12 | 9, 10, 11 | 3imtr4g 194 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 ∃wex 1381 ∈ wcel 1393 ∀wral 2306 ∃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-ial 1427 |
This theorem depends on definitions: df-bi 110 df-ral 2311 df-rex 2312 |
This theorem is referenced by: reximia 2414 reximdai 2417 r19.29 2450 reupick2 3223 ss2iun 3672 chfnrn 5278 |
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