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Mirrors > Home > ILE 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 1863 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (∃𝑦𝜑 → 𝜓)) | |
2 | 1 | albii 1450 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(∃𝑦𝜑 → 𝜓)) |
3 | 19.23v 1863 | . 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1333 ∃wex 1472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-gen 1429 ax-ie2 1474 ax-17 1506 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: ssrel 4676 ssrelrel 4688 raliunxp 4729 |
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