Theorem abeqin 34017

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 3810	. 2 ⊢ (𝐵 ∩ 𝐶) = ({𝑥 ∣ 𝜑} ∩ 𝐶)
3		abeqin.1	. 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶)
4		dfrab2 3903	. 2 ⊢ {𝑥 ∈ 𝐶 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐶)
5	2, 3, 4	3eqtr4i 2654	1 ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}

Syntax hints:   = wceq 1483  {cab 2608  {crab 2916   ∩ cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-in 3581

This theorem is referenced by:  abeqinbi  34018