Theorem r19.29r 2495

Description: Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)

Assertion

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

Proof of Theorem r19.29r

Step	Hyp	Ref	Expression
1		r19.29 2494	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
2		ancom 262	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 ∈ 𝐴 𝜑))
3		ancom 262	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
4	3	rexbii 2373	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
5	1, 2, 4	3imtr4i 199	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 102  ∀wral 2348  ∃wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by:  r19.29af2  2496