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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
StepHypRef Expression
1 sp 2171 . . 3 (∀𝑥 𝑦 = 𝑧𝑦 = 𝑧)
2 equequ2 2021 . . . . 5 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32alimi 1805 . . . 4 (∀𝑥 𝑦 = 𝑧 → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧))
4 albi 1812 . . . 4 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
53, 4syl 17 . . 3 (∀𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
61, 5jca 510 . 2 (∀𝑥 𝑦 = 𝑧 → (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
7 ax7 2011 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
87al2imi 1809 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
98a1dd 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)))
119, 10bija 379 . . 3 ((∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧) → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
1211impcom 406 . 2 ((𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
136, 12impbii 208 1 (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
Colors of variables: wff setvar class
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
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