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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 2510 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜑)) | |
2 | ancom 262 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
3 | 2 | rexbii 2373 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑)) |
4 | ancom 262 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜑)) | |
5 | 1, 3, 4 | 3bitr4i 210 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-rex 2354 |
This theorem is referenced by: ceqsrexbv 2726 ceqsrex2v 2727 2reuswapdc 2794 iunrab 3725 iunin2 3741 iundif2ss 3743 iunopab 4036 elxp2 4381 cnvuni 4539 elunirn 5426 f1oiso 5485 oprabrexex2 5777 genpdflem 6697 1idprl 6780 1idpru 6781 ltexprlemm 6790 rexuz2 8669 4fvwrd4 9150 divalgb 10325 |
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