Theorem bj-abbi2i 31964

Description: Remove dependency on ax-13 2234 from abbi2i 2725. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-abbi2i.1	⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Assertion

Ref	Expression
bj-abbi2i	⊢ 𝐴 = {𝑥 ∣ 𝜑}

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-abbi2i

Step	Hyp	Ref	Expression
1		bj-abeq2 31961	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
2		bj-abbi2i.1	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	1, 2	mpgbir 1717	1 ⊢ 𝐴 = {𝑥 ∣ 𝜑}

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606

This theorem is referenced by:  bj-abid2  31970  bj-termab  32039  bj-df-nul  32208