Theorem ss2abdv 3340

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Hypothesis

Ref	Expression
ss2abdv.1	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
ss2abdv	⊢ (φ → {x ∣ ψ} ⊆ {x ∣ χ})

Distinct variable group:   φ,x

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

Proof of Theorem ss2abdv

Step	Hyp	Ref	Expression
1		ss2abdv.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀x(ψ → χ))
3		ss2ab 3335	. 2 ⊢ ({x ∣ ψ} ⊆ {x ∣ χ} ↔ ∀x(ψ → χ))
4	2, 3	sylibr 203	1 ⊢ (φ → {x ∣ ψ} ⊆ {x ∣ χ})

Syntax hints:   → wi 4  ∀wal 1540  {cab 2339   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  ssopab2  4713  mapss  6028