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Mirrors > Home > NFE Home > Th. List > 19.23 | GIF version |
Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
19.23.1 | ⊢ Ⅎxψ |
Ref | Expression |
---|---|
19.23 | ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.23.1 | . 2 ⊢ Ⅎxψ | |
2 | 19.23t 1800 | . 2 ⊢ (Ⅎxψ → (∀x(φ → ψ) ↔ (∃xφ → ψ))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ)) |
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: 19.23h 1802 exlimi 1803 exlimd 1806 nf2 1866 19.23v 1891 pm11.53 1893 ax10-16 2190 r19.3rz 3641 ralidm 3653 |
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