Theorem abeqin 38450

Description: Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.)

Hypotheses

Ref	Expression
abeqin.1	⊢ 𝐴 = (𝐵 ∩ 𝐶)
abeqin.2	⊢ 𝐵 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
abeqin	⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem abeqin

Step	Hyp	Ref	Expression
1		abeqin.2	. . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑}
2	1	ineq1i 4168	. 2 ⊢ (𝐵 ∩ 𝐶) = ({𝑥 ∣ 𝜑} ∩ 𝐶)
3		abeqin.1	. 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶)
4		dfrab2 4272	. 2 ⊢ {𝑥 ∈ 𝐶 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐶)
5	2, 3, 4	3eqtr4i 2769	1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}

Syntax hints:   = wceq 1541  {cab 2714  {crab 3399   ∩ cin 3900

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 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-in 3908

This theorem is referenced by:  abeqinbi  38451  dfcnvrefrels3  38782  dffunsALTV  38942  dfdisjs  38967