Theorem r19.29 2644

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		pm3.2 139	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	ralimi 2570	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)))
3		rexim 2601	. . 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 124	1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wral 2485  ∃wrex 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-ral 2490  df-rex 2491

This theorem is referenced by:  r19.29r  2645  r19.29d2r  2651  r19.35-1  2657  triun  4163  ralxfrd  4517  elrnmptg  4939  fun11iun  5555  fmpt  5743  fliftfun  5878  rhmdvdsr  14012  epttop  14637  tgcnp  14756  lmtopcnp  14797  txlm  14826  metss  15041  bj-findis  16053