Theorem r19.29r 1296

Description: Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers.

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 1295	. 2 ⊢ ((∀x ∈ A ψ ∧ ∃x ∈ A φ) → ∃x ∈ A (ψ ∧ φ))
2		ancom 333	. 2 ⊢ ((∃x ∈ A φ ∧ ∀x ∈ A ψ) ↔ (∀x ∈ A ψ ∧ ∃x ∈ A φ))
3		ancom 333	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
4	3	birex 1224	. 2 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x ∈ A (ψ ∧ φ))
5	1, 2, 4	3imtr4 192	1 ⊢ ((∃x ∈ A φ ∧ ∀x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ))

Syntax hints:   → wi 2   ∧ wa 196  ∀wral 1201  ∃wrex 1202

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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