Theorem dfss3f 3266

Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)

Hypotheses

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

Assertion

Ref	Expression
dfss3f	⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B)

Proof of Theorem dfss3f

Step	Hyp	Ref	Expression
1		dfss2f.1	. . 3 ⊢ ℲxA
2		dfss2f.2	. . 3 ⊢ ℲxB
3	1, 2	dfss2f 3265	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
4		df-ral 2620	. 2 ⊢ (∀x ∈ A x ∈ B ↔ ∀x(x ∈ A → x ∈ B))
5	3, 4	bitr4i 243	1 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  Ⅎwnfc 2477  ∀wral 2615   ⊆ wss 3258

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 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  nfss  3267