Theorem ss2rabdf 45604

Description: Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024.)

Hypotheses

Ref	Expression
ss2rabdf.1	⊢ Ⅎ𝑥𝜑
ss2rabdf.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))

Assertion

Ref	Expression
ss2rabdf	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

Proof of Theorem ss2rabdf

Step	Hyp	Ref	Expression
1		ss2rabdf.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ss2rabdf.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
3	1, 2	ralrimia 3239	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4	3	ss2rabd 4010	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 396  Ⅎwnf 1790   ∈ wcel 2119  {crab 3392   ⊆ wss 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-ral 3055  df-rab 3393  df-ss 3907

This theorem is referenced by:  pimxrneun  45938  preimageiingt  47170  preimaleiinlt  47171