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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 1567 | . 2 ⊢ ((ψ ∨ ∀xφ) → ∀x(ψ ∨ φ)) |
3 | orcom 646 | . 2 ⊢ ((∀xφ ∨ ψ) ↔ (ψ ∨ ∀xφ)) | |
4 | orcom 646 | . . 3 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ)) | |
5 | 4 | albii 1356 | . 2 ⊢ (∀x(φ ∨ ψ) ↔ ∀x(ψ ∨ φ)) |
6 | 2, 3, 5 | 3imtr4i 190 | 1 ⊢ ((∀xφ ∨ ψ) → ∀x(φ ∨ ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 628 ∀wal 1240 Ⅎwnf 1346 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-gen 1335 ax-4 1397 |
This theorem depends on definitions: df-bi 110 df-nf 1347 |
This theorem is referenced by: (None) |
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