Theorem abeqinbi 37121

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 37120	. 2 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}
4		abeqinbi.3	. 2 ⊢ (𝑥 ∈ 𝐶 → (𝜑 ↔ 𝜓))
5	3, 4	rabimbieq 37119	1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {cab 2710  {crab 3433   ∩ cin 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956

This theorem is referenced by:  dfrefrels2  37383  dfcnvrefrels2  37398  dfsymrels2  37415  dftrrels2  37445