Theorem abeqin 38575

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 4156	. 2 ⊢ (𝐵 ∩ 𝐶) = ({𝑥 ∣ 𝜑} ∩ 𝐶)
3		abeqin.1	. 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶)
4		dfrab2 4260	. 2 ⊢ {𝑥 ∈ 𝐶 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐶)
5	2, 3, 4	3eqtr4i 2769	1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}

Syntax hints:   = wceq 1542  {cab 2714  {crab 3389   ∩ cin 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-in 3896

This theorem is referenced by:  abeqinbi  38576  dfcnvrefrels3  38930  dffunsALTV  39089  dfdisjs  39114