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Theorem sbcie3s 16519
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 6658 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
2 fvexd 6658 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝐹𝑤) ∈ V)
3 fvexd 6658 . . . 4 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → (𝐺𝑤) ∈ V)
4 simpllr 775 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑤))
5 fveq2 6643 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
65ad3antrrr 729 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐸𝑤) = (𝐸𝑊))
74, 6eqtrd 2856 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑊))
8 sbcie3s.a . . . . . . 7 𝐴 = (𝐸𝑊)
97, 8syl6eqr 2874 . . . . . 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 6643 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐹𝑤) = (𝐹𝑊))
1211ad3antrrr 729 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐹𝑤) = (𝐹𝑊))
1310, 12eqtrd 2856 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑊))
14 sbcie3s.b . . . . . . 7 𝐵 = (𝐹𝑊)
1513, 14syl6eqr 2874 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = 𝐵)
16 simpr 488 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑤))
17 fveq2 6643 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐺𝑤) = (𝐺𝑊))
1817ad3antrrr 729 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐺𝑤) = (𝐺𝑊))
1916, 18eqtrd 2856 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑊))
20 sbcie3s.c . . . . . . 7 𝐶 = (𝐺𝑊)
2119, 20syl6eqr 2874 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = 𝐶)
22 sbcie3s.1 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))
239, 15, 21, 22syl3anc 1368 . . . . 5 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜑𝜓))
2423bicomd 226 . . . 4 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜓𝜑))
253, 24sbcied 3791 . . 3 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → ([(𝐺𝑤) / 𝑐]𝜓𝜑))
262, 25sbcied 3791 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → ([(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
271, 26sbcied 3791 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  Vcvv 3471  [wsbc 3749  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336
This theorem is referenced by:  istrkgcb  26228  istrkgld  26231  legval  26356  istrkg2d  31944  afsval  31949
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