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Mirrors > Home > ILE Home > Th. List > 19.43 | GIF version |
Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |
Ref | Expression |
---|---|
19.43 | ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbe1 1506 | . . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
2 | hbe1 1506 | . . . 4 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓) | |
3 | 1, 2 | hbor 1557 | . . 3 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∀𝑥(∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
4 | 19.8a 1601 | . . . 4 ⊢ (𝜑 → ∃𝑥𝜑) | |
5 | 19.8a 1601 | . . . 4 ⊢ (𝜓 → ∃𝑥𝜓) | |
6 | 4, 5 | orim12i 760 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
7 | 3, 6 | exlimih 1604 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
8 | orc 713 | . . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
9 | 8 | eximi 1611 | . . 3 ⊢ (∃𝑥𝜑 → ∃𝑥(𝜑 ∨ 𝜓)) |
10 | olc 712 | . . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
11 | 10 | eximi 1611 | . . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 ∨ 𝜓)) |
12 | 9, 11 | jaoi 717 | . 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓)) |
13 | 7, 12 | impbii 126 | 1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∨ wo 709 ∃wex 1503 |
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-io 710 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: 19.44 1693 19.45 1694 19.34 1695 sborv 1902 r19.43 2648 rexun 3330 unipr 3838 uniun 3843 unopab 4097 dmun 4849 coundi 5145 coundir 5146 |
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