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Mirrors > Home > ILE Home > Th. List > r19.29af2 | GIF version |
Description: A commonly used pattern based on r19.29 2624. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
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 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | |
2 | r19.29af2.p | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | r19.29af2.1 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) | |
4 | 3 | exp31 364 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
5 | 2, 4 | ralrimi 2558 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
6 | 1, 5 | jca 306 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))) |
7 | r19.29r 2625 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒))) | |
8 | r19.29af2.c | . . 3 ⊢ Ⅎ𝑥𝜒 | |
9 | pm3.35 347 | . . . 4 ⊢ ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒) | |
10 | 9 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒)) |
11 | 8, 10 | rexlimi 2597 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒)) → 𝜒) |
12 | 6, 7, 11 | 3syl 17 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 Ⅎwnf 1470 ∈ wcel 2158 ∀wral 2465 ∃wrex 2466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-4 1520 ax-17 1536 ax-ial 1544 ax-i5r 1545 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-ral 2470 df-rex 2471 |
This theorem is referenced by: r19.29af 2628 ctiunctlemfo 12453 |
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