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Mirrors > Home > ILE Home > Th. List > 19.28h | GIF version |
Description: Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
19.28h.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
19.28h | ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26 1415 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
2 | 19.28h.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | 19.3h 1490 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
4 | 3 | anbi1i 446 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
5 | 1, 4 | bitri 182 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1287 |
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 1381 ax-gen 1383 ax-4 1445 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: nfan1 1501 aaanh 1523 exan 1628 19.28v 1828 |
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