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Mirrors > Home > ILE Home > Th. List > r19.29a | GIF version |
Description: A commonly used pattern based on r19.29 2601. (Contributed by Thierry Arnoux, 22-Nov-2017.) |
Ref | Expression |
---|---|
r19.29a.1 | ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
r19.29a.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Ref | Expression |
---|---|
r19.29a | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | r19.29a.1 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) | |
3 | r19.29a.2 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | |
4 | 1, 2, 3 | r19.29af 2605 | 1 ⊢ (𝜑 → 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: cnegexlem3 8067 cnegex 8068 modqmuladdnn0 10294 uzwodc 11959 1arith 12286 neitx 12835 |
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