Theorem r19.29 2627

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 2553	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)))
3		rexim 2584	. . 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 2468  ∃wrex 2469

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-ral 2473  df-rex 2474

This theorem is referenced by:  r19.29r  2628  r19.29d2r  2634  r19.35-1  2640  triun  4129  ralxfrd  4480  elrnmptg  4897  fun11iun  5501  fmpt  5686  fliftfun  5817  rhmdvdsr  13522  epttop  14042  tgcnp  14161  lmtopcnp  14202  txlm  14231  metss  14446  bj-findis  15184