Theorem dfss2f 3264

Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)

Hypotheses

Ref	Expression
dfss2f.1	⊢ ℲxA
dfss2f.2	⊢ ℲxB

Assertion

Ref	Expression
dfss2f	⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))

Proof of Theorem dfss2f

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3262	. 2 ⊢ (A ⊆ B ↔ ∀z(z ∈ A → z ∈ B))
2		dfss2f.1	. . . . 5 ⊢ ℲxA
3	2	nfcri 2483	. . . 4 ⊢ Ⅎx z ∈ A
4		dfss2f.2	. . . . 5 ⊢ ℲxB
5	4	nfcri 2483	. . . 4 ⊢ Ⅎx z ∈ B
6	3, 5	nfim 1813	. . 3 ⊢ Ⅎx(z ∈ A → z ∈ B)
7		nfv 1619	. . 3 ⊢ Ⅎz(x ∈ A → x ∈ B)
8		eleq1 2413	. . . 4 ⊢ (z = x → (z ∈ A ↔ x ∈ A))
9		eleq1 2413	. . . 4 ⊢ (z = x → (z ∈ B ↔ x ∈ B))
10	8, 9	imbi12d 311	. . 3 ⊢ (z = x → ((z ∈ A → z ∈ B) ↔ (x ∈ A → x ∈ B)))
11	6, 7, 10	cbval 1984	. 2 ⊢ (∀z(z ∈ A → z ∈ B) ↔ ∀x(x ∈ A → x ∈ B))
12	1, 11	bitri 240	1 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476   ⊆ wss 3257

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-nan 1288  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  dfss3f  3265  ss2ab  3334