Theorem rexanaliim 2638

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Revised by Jim Kingdon, 18-Jan-2026.)

Assertion

Ref	Expression
rexanaliim	⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

Proof of Theorem rexanaliim

Step	Hyp	Ref	Expression
1		annimim 692	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	reximi 2629	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ∃𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓))
3		rexnalim 2521	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓) → ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
4	2, 3	syl 14	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wral 2510  ∃wrex 2511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-ral 2515  df-rex 2516

This theorem is referenced by:  umgr2edg1  16063  umgr2edgneu  16066