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| 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 2661 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜑)) | |
| 2 | ancom 266 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
| 3 | 2 | rexbii 2512 | . 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 2484 |
| 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 1469 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-4 1532 ax-17 1548 ax-ial 1556 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-rex 2489 |
| This theorem is referenced by: ceqsrexbv 2903 ceqsrex2v 2904 2reuswapdc 2976 iunrab 3974 iunin2 3990 iundif2ss 3992 iunopab 4327 elxp2 4692 cnvuni 4863 elunirn 5834 f1oiso 5894 oprabrexex2 6214 genpdflem 7619 1idprl 7702 1idpru 7703 ltexprlemm 7712 rexuz2 9701 4fvwrd4 10261 divalgb 12207 |
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