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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 102 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | ralimi 2384 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 𝜑) |
3 | simpr 103 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
4 | 3 | ralimi 2384 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 𝜓) |
5 | 2, 4 | jca 290 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
6 | pm3.2 126 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
7 | 6 | ral2imi 2385 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))) |
8 | 7 | imp 115 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
9 | 5, 8 | impbii 117 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wral 2306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 |
This theorem depends on definitions: df-bi 110 df-ral 2311 |
This theorem is referenced by: r19.26-2 2442 r19.26-3 2443 ralbiim 2447 r19.27av 2448 reu8 2737 ssrab 3018 r19.28m 3311 r19.27m 3316 2ralunsn 3569 iuneq2 3673 cnvpom 4860 funco 4940 fncnv 4965 funimaexglem 4982 fnres 5015 fnopabg 5022 mpteqb 5261 eqfnfv3 5267 caoftrn 5736 iinerm 6178 rexanuz 9587 recvguniq 9593 cau3lem 9710 sqrt2irr 9878 bj-indind 10056 |
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