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 1515. (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 2450 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ (𝜑 → 𝜓))) | |
2 | pm3.35 329 | . . . 4 ⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓) | |
3 | 2 | reximi 2416 | . . 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 109 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wral 2306 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-ral 2311 df-rex 2312 |
This theorem is referenced by: r19.36av 2461 r19.37 2462 iinexgm 3908 bndndx 8180 |
Copyright terms: Public domain | W3C validator |