Theorem abeqinbi 38300

Description: Intersection with class abstraction and equivalent wff's. (Contributed by Peter Mazsa, 21-Jul-2021.)

Hypotheses

Ref	Expression
abeqinbi.1	⊢ 𝐴 = (𝐵 ∩ 𝐶)
abeqinbi.2	⊢ 𝐵 = {𝑥 ∣ 𝜑}
abeqinbi.3	⊢ (𝑥 ∈ 𝐶 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
abeqinbi	⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜓}

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem abeqinbi

Step	Hyp	Ref	Expression
1		abeqinbi.1	. . 3 ⊢ 𝐴 = (𝐵 ∩ 𝐶)
2		abeqinbi.2	. . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑}
3	1, 2	abeqin 38299	. 2 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}
4		abeqinbi.3	. 2 ⊢ (𝑥 ∈ 𝐶 → (𝜑 ↔ 𝜓))
5	3, 4	rabimbieq 38298	1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {cab 2709  {crab 3395   ∩ cin 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-in 3904

This theorem is referenced by:  dfrefrels2  38615  dfcnvrefrels2  38630  dfsymrels2  38647  dftrrels2  38681