Theorem ss2rabdv 2130

Description: Deduction of restricted abstraction subclass from implication.

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

Proof of Theorem ss2rabdv

Step	Hyp	Ref	Expression
1		ss2rabdv.1	. . 3 ⊢ ((φ ⋀ x ∈ A) → (ψ → χ))
2	1	r19.21aiva 1717	. 2 ⊢ (φ → ∀x ∈ A (ψ → χ))
3		ss2rab 2126	. 2 ⊢ ({x ∈ A∣ψ} ⊆ {x ∈ A∣χ} ↔ ∀x ∈ A (ψ → χ))
4	2, 3	sylibr 200	1 ⊢ (φ → {x ∈ A∣ψ} ⊆ {x ∈ A∣χ})

Syntax hints:   → wi 3   ⋀ wa 223   ∈ wcel 960  ∀wral 1648  {crab 1651   ⊆ wss 2050

This theorem is referenced by:  rankr1id 4707  iooss1 6374  iooss2 6375  fzss1t 6504  fzss2t 6505  clsss 7684  pjspansnt 9495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rab 1655  df-in 2054  df-ss 2056

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