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Theorem rspc2vd 26989
Description: Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class 𝐷 for the second set variable 𝑦 may depend on the first set variable 𝑥. (Contributed by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
rspc2vd.a (𝑥 = 𝐴 → (𝜃𝜒))
rspc2vd.b (𝑦 = 𝐵 → (𝜒𝜓))
rspc2vd.c (𝜑𝐴𝐶)
rspc2vd.d ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐸)
rspc2vd.e (𝜑𝐵𝐸)
Assertion
Ref Expression
rspc2vd (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃𝜓))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶   𝑦,𝐷   𝑥,𝐸   𝜑,𝑥   𝜒,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥)   𝐸(𝑦)

Proof of Theorem rspc2vd
StepHypRef Expression
1 rspc2vd.e . . 3 (𝜑𝐵𝐸)
2 rspc2vd.c . . . 4 (𝜑𝐴𝐶)
3 rspc2vd.d . . . 4 ((𝜑𝑥 = 𝐴) → 𝐷 = 𝐸)
42, 3csbied 3546 . . 3 (𝜑𝐴 / 𝑥𝐷 = 𝐸)
51, 4eleqtrrd 2707 . 2 (𝜑𝐵𝐴 / 𝑥𝐷)
6 nfcsb1v 3535 . . . . 5 𝑥𝐴 / 𝑥𝐷
7 nfv 1845 . . . . 5 𝑥𝜒
86, 7nfral 2945 . . . 4 𝑥𝑦 𝐴 / 𝑥𝐷𝜒
9 csbeq1a 3528 . . . . 5 (𝑥 = 𝐴𝐷 = 𝐴 / 𝑥𝐷)
10 rspc2vd.a . . . . 5 (𝑥 = 𝐴 → (𝜃𝜒))
119, 10raleqbidv 3146 . . . 4 (𝑥 = 𝐴 → (∀𝑦𝐷 𝜃 ↔ ∀𝑦 𝐴 / 𝑥𝐷𝜒))
128, 11rspc 3294 . . 3 (𝐴𝐶 → (∀𝑥𝐶𝑦𝐷 𝜃 → ∀𝑦 𝐴 / 𝑥𝐷𝜒))
132, 12syl 17 . 2 (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃 → ∀𝑦 𝐴 / 𝑥𝐷𝜒))
14 rspc2vd.b . . 3 (𝑦 = 𝐵 → (𝜒𝜓))
1514rspcv 3296 . 2 (𝐵𝐴 / 𝑥𝐷 → (∀𝑦 𝐴 / 𝑥𝐷𝜒𝜓))
165, 13, 15sylsyld 61 1 (𝜑 → (∀𝑥𝐶𝑦𝐷 𝜃𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  csb 3519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-v 3193  df-sbc 3423  df-csb 3520
This theorem is referenced by:  frcond1  26990
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