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Mirrors > Home > NFE Home > Th. List > 19.23vv | GIF version |
Description: Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.) |
Ref | Expression |
---|---|
19.23vv | ⊢ (∀x∀y(φ → ψ) ↔ (∃x∃yφ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.23v 1891 | . . 3 ⊢ (∀y(φ → ψ) ↔ (∃yφ → ψ)) | |
2 | 1 | albii 1566 | . 2 ⊢ (∀x∀y(φ → ψ) ↔ ∀x(∃yφ → ψ)) |
3 | 19.23v 1891 | . 2 ⊢ (∀x(∃yφ → ψ) ↔ (∃x∃yφ → ψ)) | |
4 | 2, 3 | bitri 240 | 1 ⊢ (∀x∀y(φ → ψ) ↔ (∃x∃yφ → ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 ∃wex 1541 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: ssrelk 4211 eqrelk 4212 sikexlem 4295 insklem 4304 raliunxp 4823 ssopr 4846 |
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