Theorem wl-ax11-lem6 35387

Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem6	⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑))

Distinct variable group:   𝑥,𝑢

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢)

Proof of Theorem wl-ax11-lem6

Step	Hyp	Ref	Expression
1		ax-wl-11v 35381	. . 3 ⊢ (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 → ∀𝑥∀𝑢[𝑢 / 𝑦]𝜑)
2		ax-wl-11v 35381	. . 3 ⊢ (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 → ∀𝑢∀𝑥[𝑢 / 𝑦]𝜑)
3	1, 2	impbii 212	. 2 ⊢ (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑢[𝑢 / 𝑦]𝜑)
4		nfna1 2157	. . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
5		wl-ax11-lem3 35384	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦)
6	4, 5	nfan1 2202	. . . 4 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦)
7		wl-ax11-lem5 35386	. . . . 5 ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
8	7	adantl 485	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
9	6, 8	albid 2224	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦) → (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑))
10	9	ancoms 462	. 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑥∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑))
11	3, 10	syl5bb 286	1 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1540  [wsb 2074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179  ax-13 2373  ax-wl-11v 35381

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075

This theorem is referenced by:  wl-ax11-lem10  35391