Theorem rabidd 45764

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 3444	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))
4	1, 2, 3	sylanbrc 594	1 ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ∈ wcel 2149  {crab 3423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424

This theorem is referenced by:  pimiooltgt  47315  preimageiingt  47325  preimaleiinlt  47326  sssmf  47343  fsupdm  47447  finfdm  47451