Theorem wl-aleq 37153

Description: The semantics of ∀𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)

Assertion

Ref	Expression
wl-aleq	⊢ (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))

Proof of Theorem wl-aleq

Step	Hyp	Ref	Expression
1		sp 2171	. . 3 ⊢ (∀𝑥 𝑦 = 𝑧 → 𝑦 = 𝑧)
2		equequ2 2021	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
3	2	alimi 1805	. . . 4 ⊢ (∀𝑥 𝑦 = 𝑧 → ∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
4		albi 1812	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
5	3, 4	syl 17	. . 3 ⊢ (∀𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
6	1, 5	jca 510	. 2 ⊢ (∀𝑥 𝑦 = 𝑧 → (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
7		ax7 2011	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
8	7	al2imi 1809	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
9	8	a1dd 50	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
10		axc9 2375	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
11	9, 10	bija 379	. . 3 ⊢ ((∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧) → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
12	11	impcom 406	. 2 ⊢ ((𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
13	6, 12	impbii 208	1 ⊢ (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2166  ax-13 2365

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778

This theorem is referenced by:  wl-nfeqfb  37154  wl-ax11-lem2  37204