Theorem bj-abbi2i 32776

Description: Remove dependency on ax-13 2246 from abbi2i 2738. (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 32773	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
2		bj-abbi2i.1	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	1, 2	mpgbir 1726	1 ⊢ 𝐴 = {𝑥 ∣ 𝜑}

Syntax hints:   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608

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-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

This theorem is referenced by:  bj-abid2  32782  bj-termab  32846  bj-df-nul  33017