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Theorem sbcie3s 16119
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 6364 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
2 fvexd 6364 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝐹𝑤) ∈ V)
3 fvexd 6364 . . . 4 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → (𝐺𝑤) ∈ V)
4 simpllr 817 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑤))
5 fveq2 6352 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
65ad3antrrr 768 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐸𝑤) = (𝐸𝑊))
74, 6eqtrd 2794 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = (𝐸𝑊))
8 sbcie3s.a . . . . . . 7 𝐴 = (𝐸𝑊)
97, 8syl6eqr 2812 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑎 = 𝐴)
10 simplr 809 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑤))
11 fveq2 6352 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐹𝑤) = (𝐹𝑊))
1211ad3antrrr 768 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐹𝑤) = (𝐹𝑊))
1310, 12eqtrd 2794 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = (𝐹𝑊))
14 sbcie3s.b . . . . . . 7 𝐵 = (𝐹𝑊)
1513, 14syl6eqr 2812 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑏 = 𝐵)
16 simpr 479 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑤))
17 fveq2 6352 . . . . . . . . 9 (𝑤 = 𝑊 → (𝐺𝑤) = (𝐺𝑊))
1817ad3antrrr 768 . . . . . . . 8 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝐺𝑤) = (𝐺𝑊))
1916, 18eqtrd 2794 . . . . . . 7 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = (𝐺𝑊))
20 sbcie3s.c . . . . . . 7 𝐶 = (𝐺𝑊)
2119, 20syl6eqr 2812 . . . . . 6 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → 𝑐 = 𝐶)
22 sbcie3s.1 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))
239, 15, 21, 22syl3anc 1477 . . . . 5 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜑𝜓))
2423bicomd 213 . . . 4 ((((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) ∧ 𝑐 = (𝐺𝑤)) → (𝜓𝜑))
253, 24sbcied 3613 . . 3 (((𝑤 = 𝑊𝑎 = (𝐸𝑤)) ∧ 𝑏 = (𝐹𝑤)) → ([(𝐺𝑤) / 𝑐]𝜓𝜑))
262, 25sbcied 3613 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → ([(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
271, 26sbcied 3613 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 1072   = wceq 1632  Vcvv 3340  [wsbc 3576  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  istrkgcb  25554  istrkgld  25557  legval  25678  istrkg2d  31053  afsval  31058
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