Theorem ss2rabi 1554

Description: Inference of restricted abstraction subclass from implication.

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 1553	. 2 ⊢ ({x ∈ A∣φ} ⊆ {x ∈ A∣ψ} ↔ ∀x ∈ A (φ → ψ))
2		ss2rabi.1	. 2 ⊢ (x ∈ A → (φ → ψ))
3	1, 2	mprgbir 1250	1 ⊢ {x ∈ A∣φ} ⊆ {x ∈ A∣ψ}

Syntax hints:   → wi 2   ∈ wcel 1092  {crab 1204   ⊆ wss 1487

This theorem is referenced by:  rankval3 3525  ranklon 3540

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rab 1208  df-in 1491  df-ss 1492

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