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Theorem aev 1737
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1739. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
aev (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Distinct variable group:   𝑥,𝑦

Proof of Theorem aev
Dummy variables 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbae 1650 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
2 hbae 1650 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑓𝑥 𝑥 = 𝑦)
3 ax-8 1438 . . . . 5 (𝑥 = 𝑓 → (𝑥 = 𝑦𝑓 = 𝑦))
43spimv 1736 . . . 4 (∀𝑥 𝑥 = 𝑦𝑓 = 𝑦)
52, 4alrimih 1401 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑓 𝑓 = 𝑦)
6 ax-8 1438 . . . . . . . 8 (𝑦 = 𝑢 → (𝑦 = 𝑓𝑢 = 𝑓))
7 equcomi 1635 . . . . . . . 8 (𝑢 = 𝑓𝑓 = 𝑢)
86, 7syl6 33 . . . . . . 7 (𝑦 = 𝑢 → (𝑦 = 𝑓𝑓 = 𝑢))
98spimv 1736 . . . . . 6 (∀𝑦 𝑦 = 𝑓𝑓 = 𝑢)
109alequcoms 1452 . . . . 5 (∀𝑓 𝑓 = 𝑦𝑓 = 𝑢)
1110a5i 1478 . . . 4 (∀𝑓 𝑓 = 𝑦 → ∀𝑓 𝑓 = 𝑢)
12 hbae 1650 . . . . 5 (∀𝑓 𝑓 = 𝑢 → ∀𝑣𝑓 𝑓 = 𝑢)
13 ax-8 1438 . . . . . 6 (𝑓 = 𝑣 → (𝑓 = 𝑢𝑣 = 𝑢))
1413spimv 1736 . . . . 5 (∀𝑓 𝑓 = 𝑢𝑣 = 𝑢)
1512, 14alrimih 1401 . . . 4 (∀𝑓 𝑓 = 𝑢 → ∀𝑣 𝑣 = 𝑢)
16 alequcom 1451 . . . 4 (∀𝑣 𝑣 = 𝑢 → ∀𝑢 𝑢 = 𝑣)
1711, 15, 163syl 17 . . 3 (∀𝑓 𝑓 = 𝑦 → ∀𝑢 𝑢 = 𝑣)
18 ax-8 1438 . . . 4 (𝑢 = 𝑤 → (𝑢 = 𝑣𝑤 = 𝑣))
1918spimv 1736 . . 3 (∀𝑢 𝑢 = 𝑣𝑤 = 𝑣)
205, 17, 193syl 17 . 2 (∀𝑥 𝑥 = 𝑦𝑤 = 𝑣)
211, 20alrimih 1401 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  ax16  1738  a16g  1789
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