Theorem r19.40 3137

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 487	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	reximi 3109	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜑)
3		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	reximi 3109	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
5	2, 4	jca 520	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 400  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-rex 3096

This theorem is referenced by:  rexanuz  15397  txflf  24132  metequiv2  24636  mzpcompact2lem  43409