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Mirrors > Home > ILE Home > Th. List > 19.28 | GIF version |
Description: Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
19.28.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.28 | ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26 1410 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
2 | 19.28.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | 19.3 1486 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
4 | 3 | anbi1i 445 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
5 | 1, 4 | bitri 182 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-4 1440 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: aaan 1519 |
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