Theorem ss2rabi 4077

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 4071	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
2		ss2rabi.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	mprgbir 3068	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ∈ wcel 2108  {crab 3436   ⊆ wss 3951

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-ss 3968

This theorem is referenced by:  f1ossf1o  7148  mptexgf  7242  supub  9499  suplub  9500  card2on  9594  rankval4  9907  fin1a2lem12  10451  catlid  17726  catrid  17727  gsumval2  18699  lbsextlem3  21162  psrbagsn  22087  psdmul  22170  musum  27234  ppiub  27248  umgrupgr  29120  umgrislfupgr  29140  usgruspgr  29197  usgrislfuspgr  29204  disjxwwlksn  29924  wwlksnfi  29926  disjxwwlkn  29933  clwwlknclwwlkdifnum  29999  konigsbergssiedgw  30269  omssubadd  34302  bj-unrab  36927  poimirlem26  37653  poimirlem27  37654  ssrabi  38251  lclkrs2  41542  ovolval5lem3  46669