Theorem rspct 2643

Description: A closed version of rspc 2644. (Contributed by Andrew Salmon, 6-Jun-2011.)

Hypothesis

Ref	Expression
rspct.1	⊢ Ⅎxψ

Assertion

Ref	Expression
rspct	⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ B → (∀x ∈ B φ → ψ)))

Distinct variable groups:   x,A   x,B

Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem rspct

Step	Hyp	Ref	Expression
1		df-ral 2305	. . . 4 ⊢ (∀x ∈ B φ ↔ ∀x(x ∈ B → φ))
2		eleq1 2097	. . . . . . . . . 10 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
3	2	adantr 261	. . . . . . . . 9 ⊢ ((x = A ∧ (φ ↔ ψ)) → (x ∈ B ↔ A ∈ B))
4		simpr 103	. . . . . . . . 9 ⊢ ((x = A ∧ (φ ↔ ψ)) → (φ ↔ ψ))
5	3, 4	imbi12d 223	. . . . . . . 8 ⊢ ((x = A ∧ (φ ↔ ψ)) → ((x ∈ B → φ) ↔ (A ∈ B → ψ)))
6	5	ex 108	. . . . . . 7 ⊢ (x = A → ((φ ↔ ψ) → ((x ∈ B → φ) ↔ (A ∈ B → ψ))))
7	6	a2i 11	. . . . . 6 ⊢ ((x = A → (φ ↔ ψ)) → (x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ))))
8	7	alimi 1341	. . . . 5 ⊢ (∀x(x = A → (φ ↔ ψ)) → ∀x(x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ))))
9		nfv 1418	. . . . . . 7 ⊢ Ⅎx A ∈ B
10		rspct.1	. . . . . . 7 ⊢ Ⅎxψ
11	9, 10	nfim 1461	. . . . . 6 ⊢ Ⅎx(A ∈ B → ψ)
12		nfcv 2175	. . . . . 6 ⊢ ℲxA
13	11, 12	spcgft 2624	. . . . 5 ⊢ (∀x(x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ))) → (A ∈ B → (∀x(x ∈ B → φ) → (A ∈ B → ψ))))
14	8, 13	syl 14	. . . 4 ⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ B → (∀x(x ∈ B → φ) → (A ∈ B → ψ))))
15	1, 14	syl7bi 154	. . 3 ⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ B → (∀x ∈ B φ → (A ∈ B → ψ))))
16	15	com34 77	. 2 ⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ B → (A ∈ B → (∀x ∈ B φ → ψ))))
17	16	pm2.43d 44	1 ⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ B → (∀x ∈ B φ → ψ)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  ∀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-bndl 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: (None)