Theorem r19.40 3128

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 488	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	reximi 3100	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜑)
3		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	reximi 3100	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
5	2, 4	jca 521	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-rex 3087

This theorem is used by:  rexanuz  15513  txflf  24325  metequiv2  24829  mzpcompact2lem  43762