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Mirrors > Home > ILE Home > Th. List > 19.42 | GIF version |
Description: Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
19.42.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.42 | ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.42.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | 19.41 1665 | . 2 ⊢ (∃𝑥(𝜓 ∧ 𝜑) ↔ (∃𝑥𝜓 ∧ 𝜑)) |
3 | exancom 1588 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) | |
4 | ancom 264 | . 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓 ∧ 𝜑)) | |
5 | 2, 3, 4 | 3bitr4i 211 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 Ⅎwnf 1437 ∃wex 1469 |
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 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1438 |
This theorem is referenced by: eean 1904 r2exf 2456 |
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