Theorem rexanaliim 2656

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 697	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	reximi 2647	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ∃𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓))
3		rexnalim 2539	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓) → ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
4	2, 3	syl 14	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wral 2528  ∃wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is used by:  umgr2edg1  16616  umgr2edgneu  16619