Theorem ss2rabi 4074

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 4068	. 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 3948

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 3955  df-ss 3965

This theorem is referenced by:  f1ossf1o  7123  mptexgf  7221  supub  9451  suplub  9452  card2on  9546  rankval4  9859  fin1a2lem12  10403  catlid  17624  catrid  17625  gsumval2  18602  lbsextlem3  20766  psrbagsn  21616  musum  26685  ppiub  26697  umgrupgr  28353  umgrislfupgr  28373  usgruspgr  28428  usgrislfuspgr  28434  disjxwwlksn  29148  wwlksnfi  29150  disjxwwlkn  29157  clwwlknclwwlkdifnum  29223  konigsbergssiedgw  29493  omssubadd  33288  bj-unrab  35795  poimirlem26  36503  poimirlem27  36504  ssrabi  37106  lclkrs2  40400  ovolval5lem3  45357