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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 3216. (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 423 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
6 | 2, 3, 5 | rexlimd 3276 | . 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 399 Ⅎwnf 1785 ∈ wcel 2111 ∃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: r19.29af 3289 restmetu 23177 opreu2reuALT 30247 aciunf1lem 30425 fprodex01 30567 locfinreflem 31193 esumrnmpt2 31437 esum2dlem 31461 esum2d 31462 esumiun 31463 |
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