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Theorem rspc2daf 30212
Description: Double restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
Hypotheses
Ref Expression
sbc2iedf.1 𝑥𝜑
sbc2iedf.2 𝑦𝜑
sbc2iedf.3 𝑥𝜒
sbc2iedf.4 𝑦𝜒
sbc2iedf.5 (𝜑𝐴𝑉)
sbc2iedf.6 (𝜑𝐵𝑊)
sbc2iedf.7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
rspc2daf.8 (𝜑 → ∀𝑥𝑉𝑦𝑊 𝜓)
Assertion
Ref Expression
rspc2daf (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem rspc2daf
StepHypRef Expression
1 rspc2daf.8 . . . 4 (𝜑 → ∀𝑥𝑉𝑦𝑊 𝜓)
2 sbc2iedf.1 . . . . 5 𝑥𝜑
3 nfcv 2973 . . . . . 6 𝑥𝑊
4 nfsbc1v 3768 . . . . . 6 𝑥[𝐴 / 𝑥]𝜓
53, 4nfralw 3212 . . . . 5 𝑥𝑦𝑊 [𝐴 / 𝑥]𝜓
6 sbc2iedf.5 . . . . 5 (𝜑𝐴𝑉)
7 sbc2iedf.2 . . . . . . 7 𝑦𝜑
8 nfv 1915 . . . . . . 7 𝑦 𝑥 = 𝐴
97, 8nfan 1900 . . . . . 6 𝑦(𝜑𝑥 = 𝐴)
10 sbceq1a 3759 . . . . . . 7 (𝑥 = 𝐴 → (𝜓[𝐴 / 𝑥]𝜓))
1110adantl 484 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜓[𝐴 / 𝑥]𝜓))
129, 11ralbid 3218 . . . . 5 ((𝜑𝑥 = 𝐴) → (∀𝑦𝑊 𝜓 ↔ ∀𝑦𝑊 [𝐴 / 𝑥]𝜓))
132, 5, 6, 12rspcdf 3586 . . . 4 (𝜑 → (∀𝑥𝑉𝑦𝑊 𝜓 → ∀𝑦𝑊 [𝐴 / 𝑥]𝜓))
141, 13mpd 15 . . 3 (𝜑 → ∀𝑦𝑊 [𝐴 / 𝑥]𝜓)
15 nfsbc1v 3768 . . . 4 𝑦[𝐵 / 𝑦][𝐴 / 𝑥]𝜓
16 sbc2iedf.6 . . . 4 (𝜑𝐵𝑊)
17 sbceq1a 3759 . . . . 5 (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑦][𝐴 / 𝑥]𝜓))
1817adantl 484 . . . 4 ((𝜑𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑦][𝐴 / 𝑥]𝜓))
197, 15, 16, 18rspcdf 3586 . . 3 (𝜑 → (∀𝑦𝑊 [𝐴 / 𝑥]𝜓[𝐵 / 𝑦][𝐴 / 𝑥]𝜓))
2014, 19mpd 15 . 2 (𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜓)
21 sbc2iedf.3 . . . 4 𝑥𝜒
22 sbc2iedf.4 . . . 4 𝑦𝜒
23 sbc2iedf.7 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
242, 7, 21, 22, 6, 16, 23sbc2iedf 30211 . . 3 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
25 sbccom 3828 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓[𝐵 / 𝑦][𝐴 / 𝑥]𝜓)
2624, 25bitr3di 288 . 2 (𝜑 → (𝜒[𝐵 / 𝑦][𝐴 / 𝑥]𝜓))
2720, 26mpbird 259 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wnf 1784  wcel 2114  wral 3125  [wsbc 3748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-v 3472  df-sbc 3749
This theorem is referenced by:  opreu2reuALT  30221
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