Theorem bj-rspgt 9899

Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2653 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 2100	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2	1	imbi1d 220	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) ↔ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑))))
3	2	biimpd 132	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑))))
4		imim2 49	. . . . . . . 8 ⊢ ((𝜑 → 𝜓) → ((∀𝑥 ∈ 𝐵 𝜑 → 𝜑) → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))
5	4	imim2d 48	. . . . . . 7 ⊢ ((𝜑 → 𝜓) → ((𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))))
6	3, 5	syl9 66	. . . . . 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 1344	. . . 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 2186	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
12		nfra1 2355	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐵 𝜑
13		bj-rspg.nf2	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
14	12, 13	nfim 1464	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐵 𝜑 → 𝜓)
15	11, 14	nfim 1464	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
16		rsp 2369	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐵 → 𝜑))
17	16	a1i 9	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑥 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐵 → 𝜑)))
18	17	com23 72	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜑)))
19	9, 15, 18	bj-vtoclgft 9888	. . . 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 44	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))
22	21	com23 72	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  Ⅎwnfc 2165  ∀wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559

This theorem is referenced by:  bj-rspg  9900