Theorem ss2rabdv 3683

Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)

Hypothesis

Ref	Expression
ss2rabdv.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))

Assertion

Ref	Expression
ss2rabdv	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem ss2rabdv

Step	Hyp	Ref	Expression
1		ss2rabdv.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
2	1	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
3		ss2rab 3678	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4	2, 3	sylibr 224	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by:  sess1  5082  suppfnss  7320  suppssov1  7327  suppssfv  7331  harword  8470  mrcss  16276  ablfac1b  18469  mptscmfsupp0  18928  lspss  18984  aspss  19332  dsmmacl  20085  dsmmsubg  20087  dsmmlss  20088  scmatdmat  20321  clsss  20858  lfinpfin  21327  qustgpopn  21923  metss2lem  22316  equivcau  23098  rrxmvallem  23187  ovolsslem  23252  itg2monolem1  23517  lgamucov  24764  sqff1o  24908  musum  24917  cusgrfilem1  26351  locfinreflem  29907  omsmon  30360  orvclteinc  30537  fin2solem  33395  poimirlem26  33435  poimirlem27  33436  cnambfre  33458  pclssN  35180  2polssN  35201  dihglblem3N  36584  dochss  36654  mapdordlem2  36926  nzss  38516  rmsuppss  42151  mndpsuppss  42152  scmsuppss  42153