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Mirrors > Home > MPE Home > Th. List > moanimlem | Structured version Visualization version GIF version |
Description: Factor out the common proof skeleton of moanimv 2619 and moanim 2620. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) Factor out common proof lines. (Revised by Wolf Lammen, 8-Feb-2023.) |
Ref | Expression |
---|---|
moanimlem.1 | ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓))) |
moanimlem.2 | ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
Ref | Expression |
---|---|
moanimlem | ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moanimlem.1 | . . 3 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑 ∧ 𝜓))) | |
2 | 1 | biimprcd 249 | . 2 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃*𝑥𝜓)) |
3 | moanimlem.2 | . . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) | |
4 | nexmo 2539 | . . . 4 ⊢ (¬ ∃𝑥(𝜑 ∧ 𝜓) → ∃*𝑥(𝜑 ∧ 𝜓)) | |
5 | 3, 4 | nsyl5 159 | . . 3 ⊢ (¬ 𝜑 → ∃*𝑥(𝜑 ∧ 𝜓)) |
6 | moan 2550 | . . 3 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓)) | |
7 | 5, 6 | ja 186 | . 2 ⊢ ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓)) |
8 | 2, 7 | impbii 208 | 1 ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∃wex 1781 ∃*wmo 2536 |
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 1913 ax-6 1971 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-mo 2538 |
This theorem is referenced by: moanimv 2619 moanim 2620 |
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