Theorem ss2abi 3296

Description: Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Hypothesis

Ref	Expression
ss2abi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
ss2abi	⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓}

Proof of Theorem ss2abi

Step	Hyp	Ref	Expression
1		ss2ab 3292	. 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓))
2		ss2abi.1	. 2 ⊢ (𝜑 → 𝜓)
3	1, 2	mpgbir 1499	1 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓}

Syntax hints:   → wi 4  {cab 2215   ⊆ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210

This theorem is referenced by:  abssi  3299  rabssab  3312  pwsnss  3881  iinuniss  4047  pwpwssunieq  4053  abssexg  4265  imassrn  5075  imadiflem  5396  imainlem  5398  fabexg  5509  f1oabexg  5580  tfrcllemssrecs  6488  mapex  6791  tgval  13281  tgvalex  13282  fngsum  13407  igsumvalx  13408  isghm  13766