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Mirrors > Home > ILE Home > Th. List > r19.29an | GIF version |
Description: A commonly used pattern based on r19.29 2601. (Contributed by Thierry Arnoux, 29-Dec-2019.) |
Ref | Expression |
---|---|
r19.29an.1 | ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
r19.29an | ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nfre1 2507 | . . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜓 | |
3 | 1, 2 | nfan 1552 | . 2 ⊢ Ⅎ𝑥(𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) |
4 | r19.29an.1 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) | |
5 | 4 | adantllr 473 | . 2 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
6 | simpr 109 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 𝜓) | |
7 | 3, 5, 6 | r19.29af 2605 | 1 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 ∃wrex 2443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-ral 2447 df-rex 2448 |
This theorem is referenced by: exmidontriimlem2 7170 summodclem2 11313 |
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