Theorem ss2abi 3300

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 3296	. 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓))
2		ss2abi.1	. 2 ⊢ (𝜑 → 𝜓)
3	1, 2	mpgbir 1502	1 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓}

Syntax hints:   → wi 4  {cab 2217   ⊆ wss 3201

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-in 3207  df-ss 3214

This theorem is referenced by:  abssi  3303  rabssab  3317  pwsnss  3892  iinuniss  4058  pwpwssunieq  4064  abssexg  4278  imassrn  5093  imadiflem  5416  imainlem  5418  fabexg  5532  f1oabexg  5604  tfrcllemssrecs  6561  mapex  6866  tgval  13406  tgvalex  13407  fngsum  13532  igsumvalx  13533  isghm  13891  wksfval  16243