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Mirrors > Home > ILE Home > Th. List > r19.29a | GIF version |
Description: A commonly used pattern based on r19.29 2627. (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 1539 | . 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 2631 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 ∃wrex 2469 |
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 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-ral 2473 df-rex 2474 |
This theorem is referenced by: cnegexlem3 8152 cnegex 8153 modqmuladdnn0 10386 uzwodc 12056 1arith 12383 mhmid 13023 mhmmnd 13024 ghmgrp 13026 ghmcmn 13226 ringinvnz1ne0 13362 neitx 14165 |
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