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Theorem sbcie2s 15897
Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
sbcie2s.a 𝐴 = (𝐸𝑊)
sbcie2s.b 𝐵 = (𝐹𝑊)
sbcie2s.1 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
sbcie2s (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏]𝜓𝜑))
Distinct variable groups:   𝑎,𝑏,𝑤   𝐸,𝑎,𝑏   𝐹,𝑏   𝑊,𝑎,𝑏   𝜑,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑤,𝑎,𝑏)   𝐴(𝑤,𝑎,𝑏)   𝐵(𝑤,𝑎,𝑏)   𝐸(𝑤)   𝐹(𝑤,𝑎)   𝑊(𝑤)

Proof of Theorem sbcie2s
StepHypRef Expression
1 fvex 6188 . 2 (𝐸𝑤) ∈ V
2 fvex 6188 . 2 (𝐹𝑤) ∈ V
3 simprl 793 . . . . . 6 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → 𝑎 = (𝐸𝑤))
4 fveq2 6178 . . . . . . . 8 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
5 sbcie2s.a . . . . . . . 8 𝐴 = (𝐸𝑊)
64, 5syl6eqr 2672 . . . . . . 7 (𝑤 = 𝑊 → (𝐸𝑤) = 𝐴)
76adantr 481 . . . . . 6 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → (𝐸𝑤) = 𝐴)
83, 7eqtrd 2654 . . . . 5 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → 𝑎 = 𝐴)
9 simprr 795 . . . . . 6 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → 𝑏 = (𝐹𝑤))
10 fveq2 6178 . . . . . . . 8 (𝑤 = 𝑊 → (𝐹𝑤) = (𝐹𝑊))
11 sbcie2s.b . . . . . . . 8 𝐵 = (𝐹𝑊)
1210, 11syl6eqr 2672 . . . . . . 7 (𝑤 = 𝑊 → (𝐹𝑤) = 𝐵)
1312adantr 481 . . . . . 6 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → (𝐹𝑤) = 𝐵)
149, 13eqtrd 2654 . . . . 5 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → 𝑏 = 𝐵)
15 sbcie2s.1 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜓))
168, 14, 15syl2anc 692 . . . 4 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → (𝜑𝜓))
1716bicomd 213 . . 3 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → (𝜓𝜑))
1817ex 450 . 2 (𝑤 = 𝑊 → ((𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤)) → (𝜓𝜑)))
191, 2, 18sbc2iedv 3500 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  [wsbc 3429  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884
This theorem is referenced by:  istrkgc  25334  istrkgb  25335  istrkge  25337  istrkgl  25338  ishpg  25632  iscgra  25682
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