![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > r19.35-1 | GIF version |
Description: Restricted quantifier version of 19.35-1 1604. (Contributed by Jim Kingdon, 4-Jun-2018.) |
Ref | Expression |
---|---|
r19.35-1 | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.29 2572 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ (𝜑 → 𝜓))) | |
2 | pm3.35 345 | . . . 4 ⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓) | |
3 | 2 | reximi 2532 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ (𝜑 → 𝜓)) → ∃𝑥 ∈ 𝐴 𝜓) |
4 | 1, 3 | syl 14 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) → ∃𝑥 ∈ 𝐴 𝜓) |
5 | 4 | expcom 115 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wral 2417 ∃wrex 2418 |
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 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-ral 2422 df-rex 2423 |
This theorem is referenced by: r19.36av 2585 r19.37 2586 iinexgm 4087 bndndx 9000 |
Copyright terms: Public domain | W3C validator |