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Theorem sbcie3s 17042
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 fvexd 6861 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
2 fvexd 6861 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝐹𝑤) ∈ V)
3 fvexd 6861 . . . 4 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → (𝐺𝑤) ∈ V)
4 simpllr 775 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑤))
5 fveq2 6846 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
65ad3antrrr 729 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐸𝑤) = (𝐸𝑊))
74, 6eqtrd 2773 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑊))
8 sbcie3s.a . . . . . . 7 𝐴 = (𝐸𝑊)
97, 8eqtr4di 2791 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = 𝐴)
10 simplr 768 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑤))
11 fveq2 6846 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐹𝑤) = (𝐹𝑊))
1211ad3antrrr 729 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐹𝑤) = (𝐹𝑊))
1310, 12eqtrd 2773 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑊))
14 sbcie3s.b . . . . . . 7 𝐵 = (𝐹𝑊)
1513, 14eqtr4di 2791 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = 𝐵)
16 simpr 486 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑤))
17 fveq2 6846 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐺𝑤) = (𝐺𝑊))
1817ad3antrrr 729 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐺𝑤) = (𝐺𝑊))
1916, 18eqtrd 2773 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑊))
20 sbcie3s.c . . . . . . 7 𝐶 = (𝐺𝑊)
2119, 20eqtr4di 2791 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = 𝐶)
22 sbcie3s.1 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))
239, 15, 21, 22syl3anc 1372 . . . . 5 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜑𝜓))
2423bicomd 222 . . . 4 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜓𝜑))
253, 24sbcied 3788 . . 3 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → ([(𝐺𝑤) / 𝑐]𝜓𝜑))
262, 25sbcied 3788 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → ([(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
271, 26sbcied 3788 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  Vcvv 3447  [wsbc 3743  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508
This theorem is referenced by:  istrkgcb  27447  istrkgld  27450  legval  27575  istrkg2d  33343  afsval  33348
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