Theorem ss2ab 4072

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 2905	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
2		nfab1 2905	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
3	1, 2	dfssf 3986	. 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}))
4		abid 2716	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5		abid 2716	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
6	4, 5	imbi12i 350	. . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 → 𝜓))
7	6	albii 1816	. 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 206  ∀wal 1535   ∈ wcel 2106  {cab 2712   ⊆ wss 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-clel 2814  df-nfc 2890  df-ss 3980

This theorem is referenced by:  abss  4073  ssab  4074  ss2rab  4081  rabss2  4088  rabsssn  4673  rabsspr  32529  rabsstp  32530  bj-gabss  36918  clss2lem  43601  ssabf  45040  abssf  45052  cfsetssfset  47006  sprssspr  47406