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Mirrors > Home > MPE Home > Th. List > 19.23vv | Structured version Visualization version GIF version |
Description: Theorem 19.23v 1889 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 1889 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (∃𝑦𝜑 → 𝜓)) | |
2 | 1 | albii 1737 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(∃𝑦𝜑 → 𝜓)) |
3 | 19.23v 1889 | . 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) | |
4 | 2, 3 | bitri 263 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: ssrel 5130 ssrelOLD 5131 ssrelrel 5143 raliunxp 5183 bnj1052 30297 bnj1030 30309 |
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