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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 1452 | . . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
2 | hbe1 1452 | . . . 4 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓) | |
3 | 1, 2 | hbor 1506 | . . 3 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∀𝑥(∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
4 | 19.8a 1550 | . . . 4 ⊢ (𝜑 → ∃𝑥𝜑) | |
5 | 19.8a 1550 | . . . 4 ⊢ (𝜓 → ∃𝑥𝜓) | |
6 | 4, 5 | orim12i 731 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
7 | 3, 6 | exlimih 1553 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓)) |
8 | orc 684 | . . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
9 | 8 | eximi 1560 | . . 3 ⊢ (∃𝑥𝜑 → ∃𝑥(𝜑 ∨ 𝜓)) |
10 | olc 683 | . . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
11 | 10 | eximi 1560 | . . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 ∨ 𝜓)) |
12 | 9, 11 | jaoi 688 | . 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 680 ∃wex 1449 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-4 1468 ax-ial 1495 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.44 1641 19.45 1642 19.34 1643 sborv 1842 r19.43 2561 rexun 3220 unipr 3714 uniun 3719 unopab 3965 dmun 4704 coundi 4996 coundir 4997 |
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