Theorem ss2rabi 4016

Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.) Avoid axioms. (Revised by SN, 4-Feb-2025.)

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		ss2rabi.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
2	1	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓))
3	2	ss2rabdv 4015	. 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
4	3	mptru 1549	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4  ⊤wtru 1543   ∈ wcel 2114  {crab 3389   ⊆ wss 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-ral 3052  df-rab 3390  df-ss 3906

This theorem is referenced by:  f1ossf1o  7081  mptexgf  7177  supub  9372  suplub  9373  card2on  9469  rankval4  9791  fin1a2lem12  10333  catlid  17649  catrid  17650  gsumval2  18654  lbsextlem3  21158  psrbagsn  22041  psdmul  22132  musum  27154  ppiub  27167  umgrupgr  29172  umgrislfupgr  29192  usgruspgr  29249  usgrislfuspgr  29256  disjxwwlksn  29972  wwlksnfi  29974  disjxwwlkn  29981  clwwlknclwwlkdifnum  30050  konigsbergssiedgw  30320  omssubadd  34444  bj-unrab  37233  poimirlem26  37967  poimirlem27  37968  ssrabi  38573  lclkrs2  41986  ovolval5lem3  47082