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Mirrors > Home > MPE Home > Th. List > r19.23t | Structured version Visualization version GIF version |
Description: Closed theorem form of r19.23 3273. (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 2208 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))) | |
2 | df-ral 3111 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
3 | impexp 454 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
4 | 3 | albii 1821 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) |
5 | 2, 4 | bitr4i 281 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) |
6 | df-rex 3112 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
7 | 6 | imbi1i 353 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) |
8 | 1, 5, 7 | 3bitr4g 317 | 1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-ral 3111 df-rex 3112 |
This theorem is referenced by: r19.23 3273 rexlimd2 3275 riotasv3d 36256 |
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