Theorem ss2rabi 4043

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 4037	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
2		ss2rabi.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	mprgbir 3052	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ∈ wcel 2109  {crab 3408   ⊆ wss 3917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rab 3409  df-ss 3934

This theorem is referenced by:  f1ossf1o  7103  mptexgf  7199  supub  9417  suplub  9418  card2on  9514  rankval4  9827  fin1a2lem12  10371  catlid  17651  catrid  17652  gsumval2  18620  lbsextlem3  21077  psrbagsn  21977  psdmul  22060  musum  27108  ppiub  27122  umgrupgr  29037  umgrislfupgr  29057  usgruspgr  29114  usgrislfuspgr  29121  disjxwwlksn  29841  wwlksnfi  29843  disjxwwlkn  29850  clwwlknclwwlkdifnum  29916  konigsbergssiedgw  30186  omssubadd  34298  bj-unrab  36921  poimirlem26  37647  poimirlem27  37648  ssrabi  38246  lclkrs2  41541  ovolval5lem3  46659