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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 1488 | . . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
2 | hbe1 1488 | . . . 4 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓) | |
3 | 1, 2 | hbor 1539 | . . 3 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∀𝑥(∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
4 | 19.8a 1583 | . . . 4 ⊢ (𝜑 → ∃𝑥𝜑) | |
5 | 19.8a 1583 | . . . 4 ⊢ (𝜓 → ∃𝑥𝜓) | |
6 | 4, 5 | orim12i 754 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
7 | 3, 6 | exlimih 1586 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
8 | orc 707 | . . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
9 | 8 | eximi 1593 | . . 3 ⊢ (∃𝑥𝜑 → ∃𝑥(𝜑 ∨ 𝜓)) |
10 | olc 706 | . . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
11 | 10 | eximi 1593 | . . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 ∨ 𝜓)) |
12 | 9, 11 | jaoi 711 | . 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓)) |
13 | 7, 12 | impbii 125 | 1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 703 ∃wex 1485 |
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 704 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.44 1675 19.45 1676 19.34 1677 sborv 1883 r19.43 2628 rexun 3307 unipr 3808 uniun 3813 unopab 4066 dmun 4816 coundi 5110 coundir 5111 |
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