Theorem rabeqel 38243

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 3662	. 2 ⊢ (𝐶 ∈ 𝐵 ↔ (𝐶 ∈ 𝐴 ∧ 𝜓))
4	3	biancomi 462	1 ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449

This theorem is referenced by:  elrefrels2  38509  elrefrels3  38510  elcnvrefrels2  38525  elcnvrefrels3  38526  elsymrels2  38544  elsymrels3  38545  elsymrels4  38546  elsymrels5  38547  elrefsymrels2  38560  eltrrels2  38570  eltrrels3  38571  eleqvrels2  38583  eleqvrels3  38584  elfunsALTV  38684  eldisjs  38714