Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > r19.29an | GIF version |
Description: A commonly used pattern based on r19.29 2569. (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 1508 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nfre1 2476 | . . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜓 | |
3 | 1, 2 | nfan 1544 | . 2 ⊢ Ⅎ𝑥(𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) |
4 | r19.29an.1 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) | |
5 | 4 | adantllr 472 | . 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 2573 | 1 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ∃wrex 2417 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-ral 2421 df-rex 2422 |
This theorem is referenced by: summodclem2 11151 |
Copyright terms: Public domain | W3C validator |