Theorem ss2rabi 4075

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 4069	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
2		ss2rabi.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	mprgbir 3069	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ∈ wcel 2107  {crab 3433   ⊆ wss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966

This theorem is referenced by:  f1ossf1o  7126  mptexgf  7224  supub  9454  suplub  9455  card2on  9549  rankval4  9862  fin1a2lem12  10406  catlid  17627  catrid  17628  gsumval2  18605  lbsextlem3  20773  psrbagsn  21624  musum  26695  ppiub  26707  umgrupgr  28363  umgrislfupgr  28383  usgruspgr  28438  usgrislfuspgr  28444  disjxwwlksn  29158  wwlksnfi  29160  disjxwwlkn  29167  clwwlknclwwlkdifnum  29233  konigsbergssiedgw  29503  omssubadd  33299  bj-unrab  35806  poimirlem26  36514  poimirlem27  36515  ssrabi  37117  lclkrs2  40411  ovolval5lem3  45370