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Mirrors > Home > ILE Home > Th. List > r19.23t | GIF version |
Description: Closed theorem form of r19.23 2468. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
r19.23t | ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.23t 1607 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))) | |
2 | df-ral 2353 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
3 | impexp 259 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
4 | 3 | albii 1399 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) |
5 | 2, 4 | bitr4i 185 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) |
6 | df-rex 2354 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
7 | 6 | imbi1i 236 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) |
8 | 1, 5, 7 | 3bitr4g 221 | 1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 ∃wex 1421 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-ral 2353 df-rex 2354 |
This theorem is referenced by: r19.23 2468 rexlimd2 2475 |
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