Theorem ss2rabi 3320

Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

Hypothesis

Ref	Expression
ss2rabi.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
ss2rabi	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Proof of Theorem ss2rabi

Step	Hyp	Ref	Expression
1		ss2rab 3314	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
2		ss2rabi.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	mprgbir 2600	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ∈ wcel 2203  {crab 2524   ⊆ wss 3211

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 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rab 2529  df-in 3217  df-ss 3224

This theorem is referenced by:  supubti  7290  suplubti  7291  upgruhgr  16106  umgrupgr  16107  umgrislfupgrdom  16126  uspgrushgr  16175  usgruspgr  16178  usgrislfuspgrdom  16185  konigsbergssiedgwen  16481