Theorem bj-rspgt 9194

Description: Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 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 2097	. . . . . . . . 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 1341	. . . 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 2183	. . . . . 6 ⊢ Ⅎx A ∈ B
12		nfra1 2349	. . . . . . 7 ⊢ Ⅎx∀x ∈ B φ
13		bj-rspg.nf2	. . . . . . 7 ⊢ Ⅎxψ
14	12, 13	nfim 1461	. . . . . 6 ⊢ Ⅎx(∀x ∈ B φ → ψ)
15	11, 14	nfim 1461	. . . . 5 ⊢ Ⅎx(A ∈ B → (∀x ∈ B φ → ψ))
16		rsp 2363	. . . . . . 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 9183	. . . 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 1240   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  Ⅎwnfc 2162  ∀wral 2300

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553

This theorem is referenced by:  bj-rspg  9195