Theorem ss2rabi 3349

Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

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		ss2rab 3343	. 2 ⊢ ({x ∈ A ∣ φ} ⊆ {x ∈ A ∣ ψ} ↔ ∀x ∈ A (φ → ψ))
2		ss2rabi.1	. 2 ⊢ (x ∈ A → (φ → ψ))
3	1, 2	mprgbir 2685	1 ⊢ {x ∈ A ∣ φ} ⊆ {x ∈ A ∣ ψ}

Syntax hints:   → wi 4   ∈ wcel 1710  {crab 2619   ⊆ 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-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)