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Mirrors > Home > NFE Home > Th. List > 19.21t | GIF version |
Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
Ref | Expression |
---|---|
19.21t | ⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfr 1761 | . . 3 ⊢ (Ⅎxφ → (φ → ∀xφ)) | |
2 | ax-5 1557 | . . 3 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ)) | |
3 | 1, 2 | syl9 66 | . 2 ⊢ (Ⅎxφ → (∀x(φ → ψ) → (φ → ∀xψ))) |
4 | 19.9t 1779 | . . . 4 ⊢ (Ⅎxφ → (∃xφ ↔ φ)) | |
5 | 4 | imbi1d 308 | . . 3 ⊢ (Ⅎxφ → ((∃xφ → ∀xψ) ↔ (φ → ∀xψ))) |
6 | 19.38 1794 | . . 3 ⊢ ((∃xφ → ∀xψ) → ∀x(φ → ψ)) | |
7 | 5, 6 | syl6bir 220 | . 2 ⊢ (Ⅎxφ → ((φ → ∀xψ) → ∀x(φ → ψ))) |
8 | 3, 7 | impbid 183 | 1 ⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 ∃wex 1541 Ⅎwnf 1544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-nf 1545 |
This theorem is referenced by: 19.21 1796 nfimd 1808 sbcom 2089 sbal2 2134 ax11indalem 2197 ax11inda2ALT 2198 r19.21t 2699 ceqsalt 2881 sbciegft 3076 |
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