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Theorem sbcie2s 16532
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 6676 . 2 (𝐸𝑤) ∈ V
2 fvex 6676 . 2 (𝐹𝑤) ∈ V
3 simprl 769 . . . . . 6 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → 𝑎 = (𝐸𝑤))
4 fveq2 6663 . . . . . . . 8 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
5 sbcie2s.a . . . . . . . 8 𝐴 = (𝐸𝑊)
64, 5syl6eqr 2872 . . . . . . 7 (𝑤 = 𝑊 → (𝐸𝑤) = 𝐴)
76adantr 483 . . . . . 6 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → (𝐸𝑤) = 𝐴)
83, 7eqtrd 2854 . . . . 5 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → 𝑎 = 𝐴)
9 simprr 771 . . . . . 6 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → 𝑏 = (𝐹𝑤))
10 fveq2 6663 . . . . . . . 8 (𝑤 = 𝑊 → (𝐹𝑤) = (𝐹𝑊))
11 sbcie2s.b . . . . . . . 8 𝐵 = (𝐹𝑊)
1210, 11syl6eqr 2872 . . . . . . 7 (𝑤 = 𝑊 → (𝐹𝑤) = 𝐵)
1312adantr 483 . . . . . 6 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → (𝐹𝑤) = 𝐵)
149, 13eqtrd 2854 . . . . 5 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → 𝑏 = 𝐵)
15 sbcie2s.1 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜓))
168, 14, 15syl2anc 586 . . . 4 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → (𝜑𝜓))
1716bicomd 225 . . 3 ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤))) → (𝜓𝜑))
1817ex 415 . 2 (𝑤 = 𝑊 → ((𝑎 = (𝐸𝑤) ∧ 𝑏 = (𝐹𝑤)) → (𝜓𝜑)))
191, 2, 18sbc2iedv 3849 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1531  [wsbc 3770  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  istrkgc  26232  istrkgb  26233  istrkge  26235  istrkgl  26236  ishpg  26537  iscgra  26587
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