Theorem abeqin 38903

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 4169	. 2 ⊢ (𝐵 ∩ 𝐶) = ({𝑥 ∣ 𝜑} ∩ 𝐶)
3		abeqin.1	. 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶)
4		dfrab2 4273	. 2 ⊢ {𝑥 ∈ 𝐶 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐶)
5	2, 3, 4	3eqtr4i 2796	1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}

Syntax hints:   = wceq 1570  {cab 2741  {crab 3416   ∩ cin 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-in 3912

This theorem is referenced by:  abeqinbi  38904  dfcnvrefrels3  39258  dffunsALTV  39417  dfdisjs  39442