Theorem rexlimivOLD 3183

Description: Obsolete version of rexlimiv 3146 as of 19-Dec-2024.) (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
rexlimivOLD.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
rexlimivOLD	⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimivOLD

Step	Hyp	Ref	Expression
1		rexlimivOLD.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
2	1	rgen 3061	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)
3		r19.23v 3181	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))
4	2, 3	mpbi 230	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-ral 3060  df-rex 3069

This theorem is referenced by: (None)