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Mirrors > Home > MPE Home > Th. List > 19.23vv | Structured version Visualization version GIF version |
Description: Theorem 19.23v 1934 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 1934 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (∃𝑦𝜑 → 𝜓)) | |
2 | 1 | albii 1811 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(∃𝑦𝜑 → 𝜓)) |
3 | 19.23v 1934 | . 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) | |
4 | 2, 3 | bitri 276 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 |
This theorem depends on definitions: df-bi 208 df-ex 1772 |
This theorem is referenced by: ssrel 5650 ssrelrel 5662 raliunxp 5703 bnj1052 32144 bnj1030 32156 |
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