Theorem wl-eujustlem1 37653

Description: Version of cbvexvw 2038 with references to ax-6 1968 listed as antecedents. (Contributed by Wolf Lammen, 18-Feb-2026.)

Hypothesis

Ref	Expression
wl-eujustlem1.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
wl-eujustlem1	⊢ ((∀𝑦∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∃𝑦 𝑥 = 𝑦) → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem wl-eujustlem1

Step	Hyp	Ref	Expression
1		wl-eujustlem1.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	biimpcd 249	. . . . . . . 8 ⊢ (¬ 𝜑 → (𝑥 = 𝑦 → ¬ 𝜓))
4	3	aleximi 1833	. . . . . . 7 ⊢ (∀𝑥 ¬ 𝜑 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥 ¬ 𝜓))
5		ax5e 1913	. . . . . . 7 ⊢ (∃𝑥 ¬ 𝜓 → ¬ 𝜓)
6	4, 5	syl6 35	. . . . . 6 ⊢ (∀𝑥 ¬ 𝜑 → (∃𝑥 𝑥 = 𝑦 → ¬ 𝜓))
7	6	alimdv 1917	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → (∀𝑦∃𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ 𝜓))
8	7	com12 32	. . . 4 ⊢ (∀𝑦∃𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝜑 → ∀𝑦 ¬ 𝜓))
9	2	biimprcd 250	. . . . . . . 8 ⊢ (¬ 𝜓 → (𝑥 = 𝑦 → ¬ 𝜑))
10	9	aleximi 1833	. . . . . . 7 ⊢ (∀𝑦 ¬ 𝜓 → (∃𝑦 𝑥 = 𝑦 → ∃𝑦 ¬ 𝜑))
11		ax5e 1913	. . . . . . 7 ⊢ (∃𝑦 ¬ 𝜑 → ¬ 𝜑)
12	10, 11	syl6 35	. . . . . 6 ⊢ (∀𝑦 ¬ 𝜓 → (∃𝑦 𝑥 = 𝑦 → ¬ 𝜑))
13	12	alimdv 1917	. . . . 5 ⊢ (∀𝑦 ¬ 𝜓 → (∀𝑥∃𝑦 𝑥 = 𝑦 → ∀𝑥 ¬ 𝜑))
14	13	com12 32	. . . 4 ⊢ (∀𝑥∃𝑦 𝑥 = 𝑦 → (∀𝑦 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))
15	8, 14	anbiim 641	. . 3 ⊢ ((∀𝑦∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∃𝑦 𝑥 = 𝑦) → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓))
16	15	notbid 318	. 2 ⊢ ((∀𝑦∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∃𝑦 𝑥 = 𝑦) → (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓))
17		df-ex 1781	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
18		df-ex 1781	. 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
19	16, 17, 18	3bitr4g 314	1 ⊢ ((∀𝑦∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∃𝑦 𝑥 = 𝑦) → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by: (None)