Theorem bj-abbidv 31967

Description: Remove dependency on ax-13 2234 from abbidv 2728. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-abbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bj-abbidv	⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem bj-abbidv

Step	Hyp	Ref	Expression
1		nfv 1830	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-abbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	bj-abbid 31966	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  {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

This theorem is referenced by:  bj-cdeqab  31975