Theorem rabeqel 38856

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 467	1 ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  {crab 3423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464

This theorem is referenced by:  elrefrels2  39197  elrefrels3  39198  elcnvrefrels2  39213  elcnvrefrels3  39214  elsymrels2  39236  elsymrels3  39237  elsymrels4  39238  elsymrels5  39239  elrefsymrels2  39252  eltrrels2  39262  eltrrels3  39263  eleqvrels2  39275  eleqvrels3  39276  elfunsALTV  39376  eldisjs  39418