Theorem rexanaliim 2648

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 693	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	reximi 2639	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ∃𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓))
3		rexnalim 2531	. 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 2520  ∃wrex 2521

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-ral 2525  df-rex 2526

This theorem is referenced by:  umgr2edg1  16191  umgr2edgneu  16194