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Theorem sbcie3s 15745
 Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
sbcie3s.a 𝐴 = (𝐸𝑊)
sbcie3s.b 𝐵 = (𝐹𝑊)
sbcie3s.c 𝐶 = (𝐺𝑊)
sbcie3s.1 ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))
Assertion
Ref Expression
sbcie3s (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑤   𝐸,𝑎,𝑏,𝑐   𝐹,𝑏,𝑐   𝐺,𝑐   𝑊,𝑎,𝑏,𝑐   𝜑,𝑎,𝑏,𝑐
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑤,𝑎,𝑏,𝑐)   𝐴(𝑤,𝑎,𝑏,𝑐)   𝐵(𝑤,𝑎,𝑏,𝑐)   𝐶(𝑤,𝑎,𝑏,𝑐)   𝐸(𝑤)   𝐹(𝑤,𝑎)   𝐺(𝑤,𝑎,𝑏)   𝑊(𝑤)

Proof of Theorem sbcie3s
StepHypRef Expression
1 fvex 6113 . . 3 (𝐸𝑤) ∈ V
21a1i 11 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
3 fvex 6113 . . . 4 (𝐹𝑤) ∈ V
43a1i 11 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝐹𝑤) ∈ V)
5 fvex 6113 . . . . 5 (𝐺𝑤) ∈ V
65a1i 11 . . . 4 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → (𝐺𝑤) ∈ V)
7 simpllr 795 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑤))
8 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
98ad3antrrr 762 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐸𝑤) = (𝐸𝑊))
107, 9eqtrd 2644 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑊))
11 sbcie3s.a . . . . . . 7 𝐴 = (𝐸𝑊)
1210, 11syl6eqr 2662 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = 𝐴)
13 simplr 788 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑤))
14 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐹𝑤) = (𝐹𝑊))
1514ad3antrrr 762 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐹𝑤) = (𝐹𝑊))
1613, 15eqtrd 2644 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑊))
17 sbcie3s.b . . . . . . 7 𝐵 = (𝐹𝑊)
1816, 17syl6eqr 2662 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = 𝐵)
19 simpr 476 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑤))
20 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐺𝑤) = (𝐺𝑊))
2120ad3antrrr 762 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐺𝑤) = (𝐺𝑊))
2219, 21eqtrd 2644 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑊))
23 sbcie3s.c . . . . . . 7 𝐶 = (𝐺𝑊)
2422, 23syl6eqr 2662 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = 𝐶)
25 sbcie3s.1 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))
2612, 18, 24, 25syl3anc 1318 . . . . 5 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜑𝜓))
2726bicomd 212 . . . 4 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜓𝜑))
286, 27sbcied 3439 . . 3 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → ([(𝐺𝑤) / 𝑐]𝜓𝜑))
294, 28sbcied 3439 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → ([(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
302, 29sbcied 3439 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  istrkgcb  25155  istrkgld  25158  legval  25279  istrkg2d  29997  afsval  30002
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