Theorem ss2abi 3339

Description: Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Hypothesis

Ref	Expression
ss2abi.1	⊢ (φ → ψ)

Assertion

Ref	Expression
ss2abi	⊢ {x ∣ φ} ⊆ {x ∣ ψ}

Proof of Theorem ss2abi

Step	Hyp	Ref	Expression
1		ss2ab 3335	. 2 ⊢ ({x ∣ φ} ⊆ {x ∣ ψ} ↔ ∀x(φ → ψ))
2		ss2abi.1	. 2 ⊢ (φ → ψ)
3	1, 2	mpgbir 1550	1 ⊢ {x ∣ φ} ⊆ {x ∣ ψ}

Syntax hints:   → wi 4  {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:  abssi  3342  rabssab  3353  imassrn  5010  mapsspm  6022