Theorem r19.29 2450

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 2384	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → (𝜑 ∧ 𝜓)))
3		rexim 2413	. . 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 2306  ∃wrex 2307

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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-ral 2311  df-rex 2312

This theorem is referenced by:  r19.29r  2451  r19.29d2r  2455  r19.35-1  2460  triun  3867  ralxfrd  4194  elrnmptg  4586  fun11iun  5147  fmpt  5319  fliftfun  5436  bj-findis  10104