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Theorem wl-aleq 34939
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 2181 . . 3 (∀𝑥 𝑦 = 𝑧𝑦 = 𝑧)
2 equequ2 2033 . . . . 5 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
32alimi 1813 . . . 4 (∀𝑥 𝑦 = 𝑧 → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧))
4 albi 1820 . . . 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 515 . 2 (∀𝑥 𝑦 = 𝑧 → (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
7 ax7 2023 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
87al2imi 1817 . . . . 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 2392 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
119, 10bija 385 . . 3 ((∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧) → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
1211impcom 411 . 2 ((𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
136, 12impbii 212 1 (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786
This theorem is referenced by:  wl-nfeqfb  34940  wl-ax11-lem2  34982
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