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Theorem aev-o 34535
 Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-c16 34496. Version of aev 2025 using ax-c11 34491. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
aev-o (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Distinct variable group:   𝑥,𝑦

Proof of Theorem aev-o
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbae-o 34507 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
2 hbae-o 34507 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑡𝑥 𝑥 = 𝑦)
3 ax7 1989 . . . . 5 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
43spimv 2293 . . . 4 (∀𝑥 𝑥 = 𝑦𝑡 = 𝑦)
52, 4alrimih 1791 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑡 𝑡 = 𝑦)
6 ax7 1989 . . . . . . . 8 (𝑦 = 𝑢 → (𝑦 = 𝑡𝑢 = 𝑡))
7 equcomi 1990 . . . . . . . 8 (𝑢 = 𝑡𝑡 = 𝑢)
86, 7syl6 35 . . . . . . 7 (𝑦 = 𝑢 → (𝑦 = 𝑡𝑡 = 𝑢))
98spimv 2293 . . . . . 6 (∀𝑦 𝑦 = 𝑡𝑡 = 𝑢)
109aecoms-o 34506 . . . . 5 (∀𝑡 𝑡 = 𝑦𝑡 = 𝑢)
1110axc4i-o 34502 . . . 4 (∀𝑡 𝑡 = 𝑦 → ∀𝑡 𝑡 = 𝑢)
12 hbae-o 34507 . . . . 5 (∀𝑡 𝑡 = 𝑢 → ∀𝑣𝑡 𝑡 = 𝑢)
13 ax7 1989 . . . . . 6 (𝑡 = 𝑣 → (𝑡 = 𝑢𝑣 = 𝑢))
1413spimv 2293 . . . . 5 (∀𝑡 𝑡 = 𝑢𝑣 = 𝑢)
1512, 14alrimih 1791 . . . 4 (∀𝑡 𝑡 = 𝑢 → ∀𝑣 𝑣 = 𝑢)
16 aecom-o 34505 . . . 4 (∀𝑣 𝑣 = 𝑢 → ∀𝑢 𝑢 = 𝑣)
1711, 15, 163syl 18 . . 3 (∀𝑡 𝑡 = 𝑦 → ∀𝑢 𝑢 = 𝑣)
18 ax7 1989 . . . 4 (𝑢 = 𝑤 → (𝑢 = 𝑣𝑤 = 𝑣))
1918spimv 2293 . . 3 (∀𝑢 𝑢 = 𝑣𝑤 = 𝑣)
205, 17, 193syl 18 . 2 (∀𝑥 𝑥 = 𝑦𝑤 = 𝑣)
211, 20alrimih 1791 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-11 2074  ax-12 2087  ax-13 2282  ax-c5 34487  ax-c4 34488  ax-c7 34489  ax-c10 34490  ax-c11 34491  ax-c9 34494 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by:  axc16g-o  34538
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