| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > r19.29af2 | Structured version Visualization version GIF version | ||
| Description: A commonly used pattern based on r19.29 3114. (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 419 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
| 6 | 2, 3, 5 | rexlimd 3266 | . 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 395 Ⅎwnf 1783 ∈ wcel 2108 ∃wrex 3070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-ral 3062 df-rex 3071 |
| This theorem is referenced by: r19.29af 3268 restmetu 24583 opreu2reuALT 32496 aciunf1lem 32672 fprodex01 32827 nsgqusf1olem1 33441 locfinreflem 33839 esumrnmpt2 34069 esum2dlem 34093 esum2d 34094 esumiun 34095 |
| Copyright terms: Public domain | W3C validator |