Theorem r19.40 2624

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 108	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	reximi 2567	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜑)
3		simpr 109	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	reximi 2567	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
5	2, 4	jca 304	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-ral 2453  df-rex 2454

This theorem is referenced by:  rexanuz  10952  metequiv2  13290