Theorem rspc2vd 3943

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 → (𝜃 ↔ 𝜒))
rspc2vd.b	⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))
rspc2vd.c	⊢ (𝜑 → 𝐴 ∈ 𝐶)
rspc2vd.d	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐸)
rspc2vd.e	⊢ (𝜑 → 𝐵 ∈ 𝐸)

Assertion

Ref	Expression
rspc2vd	⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → 𝜓))

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

Step	Hyp	Ref	Expression
1		rspc2vd.e	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
2		rspc2vd.c	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
3		rspc2vd.d	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐸)
4	2, 3	csbied 3930	. . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐷 = 𝐸)
5	1, 4	eleqtrrd 2829	. 2 ⊢ (𝜑 → 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐷)
6		nfcsb1v 3917	. . . . 5 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐷
7		nfv 1910	. . . . 5 ⊢ Ⅎ𝑥𝜒
8	6, 7	nfralw 3299	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒
9		csbeq1a 3906	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
10		rspc2vd.a	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜃 ↔ 𝜒))
11	9, 10	raleqbidv 3331	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐷 𝜃 ↔ ∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒))
12	8, 11	rspc 3596	. . 3 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → ∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒))
13	2, 12	syl 17	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → ∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒))
14		rspc2vd.b	. . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))
15	14	rspcv 3604	. 2 ⊢ (𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐷 → (∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒 → 𝜓))
16	5, 13, 15	sylsyld 61	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051  ⦋csb 3892

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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-v 3465  df-sbc 3777  df-csb 3893

This theorem is referenced by:  insubm  18801  frcond1  30194  frgrwopreglem4a  30238  ismntd  32855  dfmgc2lem  32866  urpropd  33099  isthincd2lem1  48382