Theorem r19.40 2763

Description: Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.40	⊢ (∃x ∈ A (φ ∧ ψ) → (∃x ∈ A φ ∧ ∃x ∈ A ψ))

Proof of Theorem r19.40

Step	Hyp	Ref	Expression
1		simpl 443	. . 3 ⊢ ((φ ∧ ψ) → φ)
2	1	reximi 2722	. 2 ⊢ (∃x ∈ A (φ ∧ ψ) → ∃x ∈ A φ)
3		simpr 447	. . 3 ⊢ ((φ ∧ ψ) → ψ)
4	3	reximi 2722	. 2 ⊢ (∃x ∈ A (φ ∧ ψ) → ∃x ∈ A ψ)
5	2, 4	jca 518	1 ⊢ (∃x ∈ A (φ ∧ ψ) → (∃x ∈ A φ ∧ ∃x ∈ A ψ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃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

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

This theorem is used by: (None)