Theorem ssrabi 36906

Description: Inference of restricted abstraction subclass from implication. (Contributed by Peter Mazsa, 26-Oct-2022.)

Hypothesis

Ref	Expression
ssrabi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
ssrabi	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Proof of Theorem ssrabi

Step	Hyp	Ref	Expression
1		ssrabi.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	ss2rabi 4067	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ∈ wcel 2106  {crab 3429   ⊆ wss 3941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rab 3430  df-v 3472  df-in 3948  df-ss 3958

This theorem is referenced by:  refrelsredund4  37291