Theorem ss2ab 4054

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 2901	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
2		nfab1 2901	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
3	1, 2	dfss2f 3970	. 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}))
4		abid 2709	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5		abid 2709	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
6	4, 5	imbi12i 350	. . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 → 𝜓))
7	6	albii 1814	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 → 𝜓))
8	3, 7	bitri 275	1 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   ∈ wcel 2099  {cab 2705   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-v 3473  df-in 3954  df-ss 3964

This theorem is referenced by:  abss  4055  ssab  4056  ss2abdvOLD  4060  ss2abiOLD  4062  ss2rab  4066  rabss2  4073  rabsssn  4671  bj-gabss  36413  clss2lem  43041  ssabf  44466  abssf  44478  cfsetssfset  46438  sprssspr  46821