Theorem rabidd 46169

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 3433	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))
4	1, 2, 3	sylanbrc 595	1 ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  {crab 3413

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414

This theorem is used by:  pimiooltgt  47719  preimageiingt  47729  preimaleiinlt  47730  sssmf  47747  fsupdm  47851  finfdm  47855