Theorem wl-cbvalnaed 32498

Description: wl-cbvalnae 32499 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)

Hypotheses

Ref	Expression
wl-cbvalnaed.1	⊢ Ⅎ𝑥𝜑
wl-cbvalnaed.2	⊢ Ⅎ𝑦𝜑
wl-cbvalnaed.3	⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))
wl-cbvalnaed.4	⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
wl-cbvalnaed.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
wl-cbvalnaed	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem wl-cbvalnaed

Step	Hyp	Ref	Expression
1		wl-cbvalnaed.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		wl-cbvalnaed.2	. . . 4 ⊢ Ⅎ𝑦𝜑
3		wl-cbvalnaed.5	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
4	1, 2, 3	wl-dral1d 32497	. . 3 ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))
5	4	imp 444	. 2 ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
6		nfnae 2306	. . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
7	1, 6	nfan 1816	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
8		wl-nfnae1 32495	. . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
9	2, 8	nfan 1816	. . 3 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
10		wl-cbvalnaed.3	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))
11	10	imp 444	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑦𝜓)
12		wl-cbvalnaed.4	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
13	12	imp 444	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜒)
14	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
15	7, 9, 11, 13, 14	cbv2 2258	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
16	5, 15	pm2.61dan 828	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  wl-cbvalnae  32499