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Theorem aev 1709
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1711. (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 1622 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
2 hbae 1622 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑓𝑥 𝑥 = 𝑦)
3 ax-8 1411 . . . . 5 (𝑥 = 𝑓 → (𝑥 = 𝑦𝑓 = 𝑦))
43spimv 1708 . . . 4 (∀𝑥 𝑥 = 𝑦𝑓 = 𝑦)
52, 4alrimih 1374 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑓 𝑓 = 𝑦)
6 ax-8 1411 . . . . . . . 8 (𝑦 = 𝑢 → (𝑦 = 𝑓𝑢 = 𝑓))
7 equcomi 1608 . . . . . . . 8 (𝑢 = 𝑓𝑓 = 𝑢)
86, 7syl6 33 . . . . . . 7 (𝑦 = 𝑢 → (𝑦 = 𝑓𝑓 = 𝑢))
98spimv 1708 . . . . . 6 (∀𝑦 𝑦 = 𝑓𝑓 = 𝑢)
109alequcoms 1425 . . . . 5 (∀𝑓 𝑓 = 𝑦𝑓 = 𝑢)
1110a5i 1451 . . . 4 (∀𝑓 𝑓 = 𝑦 → ∀𝑓 𝑓 = 𝑢)
12 hbae 1622 . . . . 5 (∀𝑓 𝑓 = 𝑢 → ∀𝑣𝑓 𝑓 = 𝑢)
13 ax-8 1411 . . . . . 6 (𝑓 = 𝑣 → (𝑓 = 𝑢𝑣 = 𝑢))
1413spimv 1708 . . . . 5 (∀𝑓 𝑓 = 𝑢𝑣 = 𝑢)
1512, 14alrimih 1374 . . . 4 (∀𝑓 𝑓 = 𝑢 → ∀𝑣 𝑣 = 𝑢)
16 alequcom 1424 . . . 4 (∀𝑣 𝑣 = 𝑢 → ∀𝑢 𝑢 = 𝑣)
1711, 15, 163syl 17 . . 3 (∀𝑓 𝑓 = 𝑦 → ∀𝑢 𝑢 = 𝑣)
18 ax-8 1411 . . . 4 (𝑢 = 𝑤 → (𝑢 = 𝑣𝑤 = 𝑣))
1918spimv 1708 . . 3 (∀𝑢 𝑢 = 𝑣𝑤 = 𝑣)
205, 17, 193syl 17 . 2 (∀𝑥 𝑥 = 𝑦𝑤 = 𝑣)
211, 20alrimih 1374 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  ax16  1710  a16g  1760
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