Theorem rabeqel 34362

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 3518	. 2 ⊢ (𝐶 ∈ 𝐵 ↔ (𝐶 ∈ 𝐴 ∧ 𝜓))
4	3	biancom 34339	1 ⊢ (𝐶 ∈ 𝐵 ↔ (𝜓 ∧ 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {crab 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353

This theorem is referenced by:  elrefrels2  34609  elrefrels3  34610  elcnvrefrels2  34622  elcnvrefrels3  34623  elsymrels2  34641  elsymrels3  34642  elsymrels4  34643  elsymrels5  34644  elrefsymrels2  34657