Theorem r19.29r 2756

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 2755	. 2 ⊢ ((∀x ∈ A ψ ∧ ∃x ∈ A φ) → ∃x ∈ A (ψ ∧ φ))
2		ancom 437	. 2 ⊢ ((∃x ∈ A φ ∧ ∀x ∈ A ψ) ↔ (∀x ∈ A ψ ∧ ∃x ∈ A φ))
3		ancom 437	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
4	3	rexbii 2640	. 2 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x ∈ A (ψ ∧ φ))
5	1, 2, 4	3imtr4i 257	1 ⊢ ((∃x ∈ A φ ∧ ∀x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wral 2615  ∃wrex 2616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is used by:  2reu5  3045