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Theorem sbcie2s 17198
Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) (Revised by SN, 2-Mar-2025.)
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]𝜑𝜓))
Distinct variable groups:   𝑎,𝑏,𝑤   𝐸,𝑎,𝑏   𝐹,𝑏   𝑊,𝑎,𝑏   𝜓,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑤,𝑎,𝑏)   𝜓(𝑤)   𝐴(𝑤,𝑎,𝑏)   𝐵(𝑤,𝑎,𝑏)   𝐸(𝑤)   𝐹(𝑤,𝑎)   𝑊(𝑤)

Proof of Theorem sbcie2s
StepHypRef Expression
1 fvex 6919 . 2 (𝐸𝑤) ∈ V
2 fvex 6919 . 2 (𝐹𝑤) ∈ V
3 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
4 sbcie2s.a . . . . . 6 𝐴 = (𝐸𝑊)
53, 4eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → (𝐸𝑤) = 𝐴)
65eqeq2d 2748 . . . 4 (𝑤 = 𝑊 → (𝑎 = (𝐸𝑤) ↔ 𝑎 = 𝐴))
76biimpd 229 . . 3 (𝑤 = 𝑊 → (𝑎 = (𝐸𝑤) → 𝑎 = 𝐴))
8 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → (𝐹𝑤) = (𝐹𝑊))
9 sbcie2s.b . . . . . 6 𝐵 = (𝐹𝑊)
108, 9eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → (𝐹𝑤) = 𝐵)
1110eqeq2d 2748 . . . 4 (𝑤 = 𝑊 → (𝑏 = (𝐹𝑤) ↔ 𝑏 = 𝐵))
1211biimpd 229 . . 3 (𝑤 = 𝑊 → (𝑏 = (𝐹𝑤) → 𝑏 = 𝐵))
13 sbcie2s.1 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜓))
1413a1i 11 . . 3 (𝑤 = 𝑊 → ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜓)))
157, 12, 14syl2and 608 . 2 (𝑤 = 𝑊 → ((𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤)) → (𝜑𝜓)))
161, 2, 15sbc2iedv 3867 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  [wsbc 3788  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569
This theorem is referenced by:  isassa  21876  istrkgc  28462  istrkgb  28463  istrkge  28465  istrkgl  28466  ishpg  28767  iscgra  28817
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