Theorem ss2ab 3335

Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)

Assertion

Ref	Expression
ss2ab	⊢ ({x ∣ φ} ⊆ {x ∣ ψ} ↔ ∀x(φ → ψ))

Proof of Theorem ss2ab

Step	Hyp	Ref	Expression
1		nfab1 2492	. . 3 ⊢ Ⅎx{x ∣ φ}
2		nfab1 2492	. . 3 ⊢ Ⅎx{x ∣ ψ}
3	1, 2	dfss2f 3265	. 2 ⊢ ({x ∣ φ} ⊆ {x ∣ ψ} ↔ ∀x(x ∈ {x ∣ φ} → x ∈ {x ∣ ψ}))
4		abid 2341	. . . 4 ⊢ (x ∈ {x ∣ φ} ↔ φ)
5		abid 2341	. . . 4 ⊢ (x ∈ {x ∣ ψ} ↔ ψ)
6	4, 5	imbi12i 316	. . 3 ⊢ ((x ∈ {x ∣ φ} → x ∈ {x ∣ ψ}) ↔ (φ → ψ))
7	6	albii 1566	. 2 ⊢ (∀x(x ∈ {x ∣ φ} → x ∈ {x ∣ ψ}) ↔ ∀x(φ → ψ))
8	3, 7	bitri 240	1 ⊢ ({x ∣ φ} ⊆ {x ∣ ψ} ↔ ∀x(φ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  {cab 2339   ⊆ 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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  abss  3336  ssab  3337  ss2abi  3339  ss2abdv  3340  ss2rab  3343  rabss2  3350