Theorem rspct 2948

Description: A closed version of rspc 2949. (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 2619	. . . 4 ⊢ (∀x ∈ B φ ↔ ∀x(x ∈ B → φ))
2		eleq1 2413	. . . . . . . . . 10 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
3	2	adantr 451	. . . . . . . . 9 ⊢ ((x = A ∧ (φ ↔ ψ)) → (x ∈ B ↔ A ∈ B))
4		simpr 447	. . . . . . . . 9 ⊢ ((x = A ∧ (φ ↔ ψ)) → (φ ↔ ψ))
5	3, 4	imbi12d 311	. . . . . . . 8 ⊢ ((x = A ∧ (φ ↔ ψ)) → ((x ∈ B → φ) ↔ (A ∈ B → ψ)))
6	5	ex 423	. . . . . . 7 ⊢ (x = A → ((φ ↔ ψ) → ((x ∈ B → φ) ↔ (A ∈ B → ψ))))
7	6	a2i 12	. . . . . 6 ⊢ ((x = A → (φ ↔ ψ)) → (x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ))))
8	7	alimi 1559	. . . . 5 ⊢ (∀x(x = A → (φ ↔ ψ)) → ∀x(x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ))))
9		nfv 1619	. . . . . . 7 ⊢ Ⅎx A ∈ B
10		rspct.1	. . . . . . 7 ⊢ Ⅎxψ
11	9, 10	nfim 1813	. . . . . 6 ⊢ Ⅎx(A ∈ B → ψ)
12		nfcv 2489	. . . . . 6 ⊢ ℲxA
13	11, 12	spcgft 2931	. . . . 5 ⊢ (∀x(x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ))) → (A ∈ B → (∀x(x ∈ B → φ) → (A ∈ B → ψ))))
14	8, 13	syl 15	. . . 4 ⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ B → (∀x(x ∈ B → φ) → (A ∈ B → ψ))))
15	1, 14	syl7bi 221	. . 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   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861

This theorem is referenced by: (None)