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Mirrors > Home > ILE Home > Th. List > albi | GIF version |
Description: Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
albi | ⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi1 111 | . . 3 ⊢ ((φ ↔ ψ) → (φ → ψ)) | |
2 | 1 | al2imi 1344 | . 2 ⊢ (∀x(φ ↔ ψ) → (∀xφ → ∀xψ)) |
3 | bi2 121 | . . 3 ⊢ ((φ ↔ ψ) → (ψ → φ)) | |
4 | 3 | al2imi 1344 | . 2 ⊢ (∀x(φ ↔ ψ) → (∀xψ → ∀xφ)) |
5 | 2, 4 | impbid 120 | 1 ⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 |
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-5 1333 ax-gen 1335 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: albii 1356 albidh 1366 19.16 1444 19.17 1445 intmin4 3634 dfiin2g 3681 |
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