Theorem rabidd 45585

Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by Glauco Siliprandi, 24-Jan-2025.)

Hypotheses

Ref	Expression
rabidd.1	⊢ (𝜑 → 𝑥 ∈ 𝐴)
rabidd.2	⊢ (𝜑 → 𝜒)

Assertion

Ref	Expression
rabidd	⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒})

Proof of Theorem rabidd

Step	Hyp	Ref	Expression
1		rabidd.1	. 2 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
2		rabidd.2	. 2 ⊢ (𝜑 → 𝜒)
3		rabid 3410	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))
4	1, 2, 3	sylanbrc 584	1 ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ∈ wcel 2114  {crab 3389

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390

This theorem is referenced by:  pimiooltgt  47138  preimageiingt  47148  preimaleiinlt  47149  fsupdm  47270  finfdm  47274