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Mirrors > Home > MPE Home > Th. List > r19.29af2 | Structured version Visualization version GIF version |
Description: A commonly used pattern based on r19.29 3054. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Ref | Expression |
---|---|
r19.29af2.p | ⊢ Ⅎ𝑥𝜑 |
r19.29af2.c | ⊢ Ⅎ𝑥𝜒 |
r19.29af2.1 | ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
r19.29af2.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Ref | Expression |
---|---|
r19.29af2 | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.29af2.2 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | |
2 | r19.29af2.p | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | r19.29af2.c | . . 3 ⊢ Ⅎ𝑥𝜒 | |
4 | r19.29af2.1 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) | |
5 | 4 | exp31 628 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
6 | 2, 3, 5 | rexlimd 3008 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
7 | 1, 6 | mpd 15 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 Ⅎwnf 1699 ∈ wcel 1977 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 df-nf 1701 df-ral 2901 df-rex 2902 |
This theorem is referenced by: r19.29af 3058 restmetu 22185 aciunf1lem 28844 locfinreflem 29235 esumrnmpt2 29457 esum2dlem 29481 esum2d 29482 esumiun 29483 |
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