Theorem r19.29 2444

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.)

Assertion

Ref	Expression
r19.29	⊢ ((∀x ∈ A φ ∧ ∃x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ))

Proof of Theorem r19.29

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))

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  9409