Theorem ss2ab 4017

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 2929	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
2		nfab1 2929	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
3	1, 2	dfssf 3930	. 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}))
4		abid 2747	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5		abid 2747	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
6	4, 5	imbi12i 353	. . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 → 𝜓))
7	6	albii 1842	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 → 𝜓))
8	3, 7	bitri 278	1 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   ∈ wcel 2145  {cab 2743   ⊆ wss 3907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-10 2178  ax-11 2194  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-clel 2840  df-nfc 2914  df-ss 3924

This theorem is referenced by:  abss  4018  ssab  4019  ss2rab  4025  rabss2OLD  4034  rabsssn  4630  rabsspr  32757  rabsstp  32758  bj-gabss  37432  clss2lem  44199  ssabf  45676  abssf  45688  cfsetssfset  47648  sprssspr  48085