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Mirrors > Home > ILE Home > Th. List > r19.29 | GIF version |
Description: Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
r19.29 | ⊢ ((∀x ∈ A φ ∧ ∃x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.2 126 | . . . 4 ⊢ (φ → (ψ → (φ ∧ ψ))) | |
2 | 1 | ralimi 2378 | . . 3 ⊢ (∀x ∈ A φ → ∀x ∈ A (ψ → (φ ∧ ψ))) |
3 | rexim 2407 | . . 3 ⊢ (∀x ∈ A (ψ → (φ ∧ ψ)) → (∃x ∈ A ψ → ∃x ∈ A (φ ∧ ψ))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (∀x ∈ A φ → (∃x ∈ A ψ → ∃x ∈ A (φ ∧ ψ))) |
5 | 4 | imp 115 | 1 ⊢ ((∀x ∈ A φ ∧ ∃x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wral 2300 ∃wrex 2301 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-ral 2305 df-rex 2306 |
This theorem is referenced by: r19.29r 2445 r19.29d2r 2449 r19.35-1 2454 triun 3858 ralxfrd 4160 elrnmptg 4529 fun11iun 5090 fmpt 5262 fliftfun 5379 bj-findis 9439 |
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