Theorem bj-rspgt 12982

Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2781 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
bj-rspg.nfa	⊢ Ⅎ𝑥𝐴
bj-rspg.nfb	⊢ Ⅎ𝑥𝐵
bj-rspg.nf2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
bj-rspgt	⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))

Proof of Theorem bj-rspgt

Step	Hyp	Ref	Expression
1		eleq1 2200	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	imbi1d 230	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) ↔ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑))))
3	2	biimpd 143	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑))))
4		imim2 55	. . . . . . . 8 ⊢ ((𝜑 → 𝜓) → ((∀𝑥 ∈ 𝐵 𝜑 → 𝜑) → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))
5	4	imim2d 54	. . . . . . 7 ⊢ ((𝜑 → 𝜓) → ((𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))))
6	3, 5	syl9 72	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜓) → ((𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))))
7	6	a2i 11	. . . . 5 ⊢ ((𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))))
8	7	alimi 1431	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))))
9		bj-rspg.nfa	. . . . 5 ⊢ Ⅎ𝑥𝐴
10		bj-rspg.nfb	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
11	9, 10	nfel 2288	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
12		nfra1 2464	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐵 𝜑
13		bj-rspg.nf2	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
14	12, 13	nfim 1551	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐵 𝜑 → 𝜓)
15	11, 14	nfim 1551	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
16		rsp 2478	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐵 → 𝜑))
17	16	a1i 9	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑥 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐵 → 𝜑)))
18	17	com23 78	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)))
19	9, 15, 18	bj-vtoclgft 12971	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))))
20	8, 19	syl 14	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))))
21	20	pm2.43d 50	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))
22	21	com23 78	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1329   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  Ⅎwnfc 2266  ∀wral 2414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683

This theorem is referenced by:  bj-rspg  12983