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Mirrors > Home > NFE Home > Th. List > r19.28zv | GIF version |
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.) |
Ref | Expression |
---|---|
r19.28zv | ⊢ (A ≠ ∅ → (∀x ∈ A (φ ∧ ψ) ↔ (φ ∧ ∀x ∈ A ψ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.3rzv 3643 | . . 3 ⊢ (A ≠ ∅ → (φ ↔ ∀x ∈ A φ)) | |
2 | 1 | anbi1d 685 | . 2 ⊢ (A ≠ ∅ → ((φ ∧ ∀x ∈ A ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ))) |
3 | r19.26 2746 | . 2 ⊢ (∀x ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ψ)) | |
4 | 2, 3 | syl6rbbr 255 | 1 ⊢ (A ≠ ∅ → (∀x ∈ A (φ ∧ ψ) ↔ (φ ∧ ∀x ∈ A ψ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ≠ wne 2516 ∀wral 2614 ∅c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: raaanv 3658 iinrab 4028 iindif2 4035 iinin2 4036 fint 5245 |
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