![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > r19.42v | GIF version |
Description: Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
r19.42v | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.41v 2633 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜑)) | |
2 | ancom 266 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
3 | 2 | rexbii 2484 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑)) |
4 | ancom 266 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜑)) | |
5 | 1, 3, 4 | 3bitr4i 212 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∃wrex 2456 |
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 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-rex 2461 |
This theorem is referenced by: ceqsrexbv 2868 ceqsrex2v 2869 2reuswapdc 2941 iunrab 3934 iunin2 3950 iundif2ss 3952 iunopab 4281 elxp2 4644 cnvuni 4813 elunirn 5766 f1oiso 5826 oprabrexex2 6130 genpdflem 7505 1idprl 7588 1idpru 7589 ltexprlemm 7598 rexuz2 9579 4fvwrd4 10137 divalgb 11924 |
Copyright terms: Public domain | W3C validator |