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Mirrors > Home > ILE Home > Th. List > r19.26 | GIF version |
Description: Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
r19.26 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | ralimi 2498 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 𝜑) |
3 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
4 | 3 | ralimi 2498 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 𝜓) |
5 | 2, 4 | jca 304 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
6 | pm3.2 138 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
7 | 6 | ral2imi 2500 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))) |
8 | 7 | imp 123 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
9 | 5, 8 | impbii 125 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wral 2417 |
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 1424 ax-gen 1426 |
This theorem depends on definitions: df-bi 116 df-ral 2422 |
This theorem is referenced by: r19.27v 2562 r19.28v 2563 r19.26-2 2564 r19.26-3 2565 ralbiim 2569 r19.27av 2570 reu8 2884 ssrab 3180 r19.28m 3457 r19.27m 3463 2ralunsn 3733 iuneq2 3837 cnvpom 5089 funco 5171 fncnv 5197 funimaexglem 5214 fnres 5247 fnopabg 5254 mpteqb 5519 eqfnfv3 5528 caoftrn 6015 iinerm 6509 ixpeq2 6614 ixpin 6625 rexanuz 10792 recvguniq 10799 cau3lem 10918 rexanre 11024 bezoutlemmo 11730 sqrt2irr 11876 tgval2 12259 metequiv 12703 metequiv2 12704 mulcncflem 12798 bj-indind 13301 |
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