MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcie3s Structured version   Visualization version   GIF version

Theorem sbcie3s 15964
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 6241 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
2 fvexd 6241 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝐹𝑤) ∈ V)
3 fvexd 6241 . . . 4 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → (𝐺𝑤) ∈ V)
4 simpllr 815 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑤))
5 fveq2 6229 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
65ad3antrrr 766 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐸𝑤) = (𝐸𝑊))
74, 6eqtrd 2685 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑊))
8 sbcie3s.a . . . . . . 7 𝐴 = (𝐸𝑊)
97, 8syl6eqr 2703 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = 𝐴)
10 simplr 807 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑤))
11 fveq2 6229 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐹𝑤) = (𝐹𝑊))
1211ad3antrrr 766 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐹𝑤) = (𝐹𝑊))
1310, 12eqtrd 2685 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑊))
14 sbcie3s.b . . . . . . 7 𝐵 = (𝐹𝑊)
1513, 14syl6eqr 2703 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = 𝐵)
16 simpr 476 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑤))
17 fveq2 6229 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐺𝑤) = (𝐺𝑊))
1817ad3antrrr 766 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐺𝑤) = (𝐺𝑊))
1916, 18eqtrd 2685 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑊))
20 sbcie3s.c . . . . . . 7 𝐶 = (𝐺𝑊)
2119, 20syl6eqr 2703 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = 𝐶)
22 sbcie3s.1 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))
239, 15, 21, 22syl3anc 1366 . . . . 5 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜑𝜓))
2423bicomd 213 . . . 4 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜓𝜑))
253, 24sbcied 3505 . . 3 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → ([(𝐺𝑤) / 𝑐]𝜓𝜑))
262, 25sbcied 3505 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → ([(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
271, 26sbcied 3505 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  Vcvv 3231  [wsbc 3468  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934
This theorem is referenced by:  istrkgcb  25400  istrkgld  25403  legval  25524  istrkg2d  30872  afsval  30877
  Copyright terms: Public domain W3C validator