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Mirrors > Home > ILE Home > Th. List > 19.31r | GIF version |
Description: One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Ref | Expression |
---|---|
19.31r.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
19.31r | ⊢ ((∀𝑥𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.31r.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | 19.32r 1667 | . 2 ⊢ ((𝜓 ∨ ∀𝑥𝜑) → ∀𝑥(𝜓 ∨ 𝜑)) |
3 | orcom 718 | . 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑)) | |
4 | orcom 718 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
5 | 4 | albii 1457 | . 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑)) |
6 | 2, 3, 5 | 3imtr4i 200 | 1 ⊢ ((∀𝑥𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 ∀wal 1340 Ⅎwnf 1447 |
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 1434 ax-gen 1436 ax-4 1497 |
This theorem depends on definitions: df-bi 116 df-nf 1448 |
This theorem is referenced by: (None) |
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