Theorem ss2rabi 4087

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 4081	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
2		ss2rabi.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	mprgbir 3066	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ∈ wcel 2106  {crab 3433   ⊆ 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-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

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-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-ss 3980

This theorem is referenced by:  f1ossf1o  7148  mptexgf  7242  supub  9497  suplub  9498  card2on  9592  rankval4  9905  fin1a2lem12  10449  catlid  17728  catrid  17729  gsumval2  18712  lbsextlem3  21180  psrbagsn  22105  psdmul  22188  musum  27249  ppiub  27263  umgrupgr  29135  umgrislfupgr  29155  usgruspgr  29212  usgrislfuspgr  29219  disjxwwlksn  29934  wwlksnfi  29936  disjxwwlkn  29943  clwwlknclwwlkdifnum  30009  konigsbergssiedgw  30279  omssubadd  34282  bj-unrab  36909  poimirlem26  37633  poimirlem27  37634  ssrabi  38232  lclkrs2  41523  ovolval5lem3  46610