Theorem rabeqel 36382

Description: Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021.)

Hypotheses

Ref	Expression
rabeqel.1	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
rabeqel.2	⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rabeqel	⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem rabeqel

Step	Hyp	Ref	Expression
1		rabeqel.2	. . 3 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓))
2		rabeqel.1	. . 3 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
3	1, 2	elrab2 3629	. 2 ⊢ (𝐶 ∈ 𝐵 ↔ (𝐶 ∈ 𝐴 ∧ 𝜓))
4	3	biancomi 463	1 ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  {crab 3070

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433

This theorem is referenced by:  elrefrels2  36623  elrefrels3  36624  elcnvrefrels2  36636  elcnvrefrels3  36637  elsymrels2  36655  elsymrels3  36656  elsymrels4  36657  elsymrels5  36658  elrefsymrels2  36671  eltrrels2  36681  eltrrels3  36682  eleqvrels2  36693  eleqvrels3  36694  elfunsALTV  36791  eldisjs  36821