Theorem rspc2daf 30231

Description: Double restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)

Hypotheses

Ref	Expression
sbc2iedf.1	⊢ Ⅎ𝑥𝜑
sbc2iedf.2	⊢ Ⅎ𝑦𝜑
sbc2iedf.3	⊢ Ⅎ𝑥𝜒
sbc2iedf.4	⊢ Ⅎ𝑦𝜒
sbc2iedf.5	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sbc2iedf.6	⊢ (𝜑 → 𝐵 ∈ 𝑊)
sbc2iedf.7	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
rspc2daf.8	⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜓)

Assertion

Ref	Expression
rspc2daf	⊢ (𝜑 → 𝜒)

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

Step	Hyp	Ref	Expression
1		rspc2daf.8	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜓)
2		sbc2iedf.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝑊
4		nfsbc1v 3792	. . . . . 6 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜓
5	3, 4	nfralw 3225	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑊 [𝐴 / 𝑥]𝜓
6		sbc2iedf.5	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		sbc2iedf.2	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
8		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = 𝐴
9	7, 8	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
10		sbceq1a 3783	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ [𝐴 / 𝑥]𝜓))
11	10	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ [𝐴 / 𝑥]𝜓))
12	9, 11	ralbid 3231	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (∀𝑦 ∈ 𝑊 𝜓 ↔ ∀𝑦 ∈ 𝑊 [𝐴 / 𝑥]𝜓))
13	2, 5, 6, 12	rspcdf 3610	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜓 → ∀𝑦 ∈ 𝑊 [𝐴 / 𝑥]𝜓))
14	1, 13	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑊 [𝐴 / 𝑥]𝜓)
15		nfsbc1v 3792	. . . 4 ⊢ Ⅎ𝑦[𝐵 / 𝑦][𝐴 / 𝑥]𝜓
16		sbc2iedf.6	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
17		sbceq1a 3783	. . . . 5 ⊢ (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜓))
18	17	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜓))
19	7, 15, 16, 18	rspcdf 3610	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑊 [𝐴 / 𝑥]𝜓 → [𝐵 / 𝑦][𝐴 / 𝑥]𝜓))
20	14, 19	mpd 15	. 2 ⊢ (𝜑 → [𝐵 / 𝑦][𝐴 / 𝑥]𝜓)
21		sbccom 3854	. . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜓)
22		sbc2iedf.3	. . . 4 ⊢ Ⅎ𝑥𝜒
23		sbc2iedf.4	. . . 4 ⊢ Ⅎ𝑦𝜒
24		sbc2iedf.7	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
25	2, 7, 22, 23, 6, 16, 24	sbc2iedf 30230	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
26	21, 25	syl5rbbr 288	. 2 ⊢ (𝜑 → (𝜒 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜓))
27	20, 26	mpbird 259	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∀wral 3138  [wsbc 3772

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 2793

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-sbc 3773

This theorem is referenced by:  opreu2reuALT  30240