Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > r19.29 | GIF version |
Description: Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
r19.29 | ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.2 138 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
2 | 1 | ralimi 2495 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓))) |
3 | rexim 2526 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))) |
5 | 4 | imp 123 | 1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wral 2416 ∃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-ial 1514 |
This theorem depends on definitions: df-bi 116 df-ral 2421 df-rex 2422 |
This theorem is referenced by: r19.29r 2570 r19.29d2r 2576 r19.35-1 2581 triun 4039 ralxfrd 4383 elrnmptg 4791 fun11iun 5388 fmpt 5570 fliftfun 5697 epttop 12259 tgcnp 12378 lmtopcnp 12419 txlm 12448 metss 12663 bj-findis 13177 |
Copyright terms: Public domain | W3C validator |