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Theorem exmidontriimlem4 7172
Description: Lemma for exmidontriim 7173. The induction step for the induction on 𝐴. (Contributed by Jim Kingdon, 10-Aug-2024.)
Hypotheses
Ref Expression
exmidontriimlem4.a (𝜑𝐴 ∈ On)
exmidontriimlem4.b (𝜑𝐵 ∈ On)
exmidontriimlem4.em (𝜑EXMID)
exmidontriimlem4.h (𝜑 → ∀𝑧𝐴𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))
Assertion
Ref Expression
exmidontriimlem4 (𝜑 → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Distinct variable group:   𝑦,𝐴,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑦,𝑧)

Proof of Theorem exmidontriimlem4
Dummy variables 𝑏 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2228 . . 3 (𝑏 = 𝐵 → (𝐴𝑏𝐴𝐵))
2 eqeq2 2174 . . 3 (𝑏 = 𝐵 → (𝐴 = 𝑏𝐴 = 𝐵))
3 eleq1 2227 . . 3 (𝑏 = 𝐵 → (𝑏𝐴𝐵𝐴))
41, 2, 33orbi123d 1300 . 2 (𝑏 = 𝐵 → ((𝐴𝑏𝐴 = 𝑏𝑏𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
5 eleq2w 2226 . . . . . . 7 (𝑏 = 𝑤 → (𝐴𝑏𝐴𝑤))
6 eqeq2 2174 . . . . . . 7 (𝑏 = 𝑤 → (𝐴 = 𝑏𝐴 = 𝑤))
7 eleq1w 2225 . . . . . . 7 (𝑏 = 𝑤 → (𝑏𝐴𝑤𝐴))
85, 6, 73orbi123d 1300 . . . . . 6 (𝑏 = 𝑤 → ((𝐴𝑏𝐴 = 𝑏𝑏𝐴) ↔ (𝐴𝑤𝐴 = 𝑤𝑤𝐴)))
98imbi2d 229 . . . . 5 (𝑏 = 𝑤 → ((𝜑 → (𝐴𝑏𝐴 = 𝑏𝑏𝐴)) ↔ (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))))
10 exmidontriimlem4.a . . . . . . . 8 (𝜑𝐴 ∈ On)
1110adantl 275 . . . . . . 7 (((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) → 𝐴 ∈ On)
12 simpll 519 . . . . . . 7 (((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) → 𝑏 ∈ On)
13 exmidontriimlem4.em . . . . . . . 8 (𝜑EXMID)
1413adantl 275 . . . . . . 7 (((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) → EXMID)
15 exmidontriimlem4.h . . . . . . . 8 (𝜑 → ∀𝑧𝐴𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))
1615adantl 275 . . . . . . 7 (((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) → ∀𝑧𝐴𝑦 ∈ On (𝑧𝑦𝑧 = 𝑦𝑦𝑧))
17 simplr 520 . . . . . . . . . 10 ((((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) ∧ 𝑣𝑏) → 𝜑)
18 eleq2w 2226 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → (𝐴𝑤𝐴𝑣))
19 eqeq2 2174 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → (𝐴 = 𝑤𝐴 = 𝑣))
20 eleq1w 2225 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → (𝑤𝐴𝑣𝐴))
2118, 19, 203orbi123d 1300 . . . . . . . . . . . 12 (𝑤 = 𝑣 → ((𝐴𝑤𝐴 = 𝑤𝑤𝐴) ↔ (𝐴𝑣𝐴 = 𝑣𝑣𝐴)))
2221imbi2d 229 . . . . . . . . . . 11 (𝑤 = 𝑣 → ((𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴)) ↔ (𝜑 → (𝐴𝑣𝐴 = 𝑣𝑣𝐴))))
23 simpllr 524 . . . . . . . . . . 11 ((((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) ∧ 𝑣𝑏) → ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴)))
24 simpr 109 . . . . . . . . . . 11 ((((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) ∧ 𝑣𝑏) → 𝑣𝑏)
2522, 23, 24rspcdva 2831 . . . . . . . . . 10 ((((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) ∧ 𝑣𝑏) → (𝜑 → (𝐴𝑣𝐴 = 𝑣𝑣𝐴)))
2617, 25mpd 13 . . . . . . . . 9 ((((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) ∧ 𝑣𝑏) → (𝐴𝑣𝐴 = 𝑣𝑣𝐴))
2726ralrimiva 2537 . . . . . . . 8 (((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) → ∀𝑣𝑏 (𝐴𝑣𝐴 = 𝑣𝑣𝐴))
28 eleq2w 2226 . . . . . . . . . 10 (𝑣 = 𝑦 → (𝐴𝑣𝐴𝑦))
29 eqeq2 2174 . . . . . . . . . 10 (𝑣 = 𝑦 → (𝐴 = 𝑣𝐴 = 𝑦))
30 eleq1w 2225 . . . . . . . . . 10 (𝑣 = 𝑦 → (𝑣𝐴𝑦𝐴))
3128, 29, 303orbi123d 1300 . . . . . . . . 9 (𝑣 = 𝑦 → ((𝐴𝑣𝐴 = 𝑣𝑣𝐴) ↔ (𝐴𝑦𝐴 = 𝑦𝑦𝐴)))
3231cbvralv 2690 . . . . . . . 8 (∀𝑣𝑏 (𝐴𝑣𝐴 = 𝑣𝑣𝐴) ↔ ∀𝑦𝑏 (𝐴𝑦𝐴 = 𝑦𝑦𝐴))
3327, 32sylib 121 . . . . . . 7 (((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) → ∀𝑦𝑏 (𝐴𝑦𝐴 = 𝑦𝑦𝐴))
3411, 12, 14, 16, 33exmidontriimlem3 7171 . . . . . 6 (((𝑏 ∈ On ∧ ∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴))) ∧ 𝜑) → (𝐴𝑏𝐴 = 𝑏𝑏𝐴))
3534exp31 362 . . . . 5 (𝑏 ∈ On → (∀𝑤𝑏 (𝜑 → (𝐴𝑤𝐴 = 𝑤𝑤𝐴)) → (𝜑 → (𝐴𝑏𝐴 = 𝑏𝑏𝐴))))
369, 35tfis2 4557 . . . 4 (𝑏 ∈ On → (𝜑 → (𝐴𝑏𝐴 = 𝑏𝑏𝐴)))
3736impcom 124 . . 3 ((𝜑𝑏 ∈ On) → (𝐴𝑏𝐴 = 𝑏𝑏𝐴))
3837ralrimiva 2537 . 2 (𝜑 → ∀𝑏 ∈ On (𝐴𝑏𝐴 = 𝑏𝑏𝐴))
39 exmidontriimlem4.b . 2 (𝜑𝐵 ∈ On)
404, 38, 39rspcdva 2831 1 (𝜑 → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3o 966   = wceq 1342  wcel 2135  wral 2442  EXMIDwem 4168  Oncon0 4336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-setind 4509
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2724  df-dif 3114  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-uni 3785  df-tr 4076  df-exmid 4169  df-iord 4339  df-on 4341
This theorem is referenced by:  exmidontriim  7173
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