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Mirrors > Home > ILE Home > Th. List > 19.45 | GIF version |
Description: Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Ref | Expression |
---|---|
19.45.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.45 | ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.43 1608 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) | |
2 | 19.45.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | 19.9 1624 | . . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
4 | 3 | orbi1i 753 | . 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) |
5 | 1, 4 | bitri 183 | 1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 698 Ⅎwnf 1440 ∃wex 1472 |
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-io 699 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: eeor 1675 |
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