Theorem bj-abbid 32778

Description: Remove dependency on ax-13 2246 from abbid 2740. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-abbid.1	⊢ Ⅎ𝑥𝜑
bj-abbid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem bj-abbid

Step	Hyp	Ref	Expression
1		bj-abbid.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		bj-abbid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	alrimi 2082	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		bj-abbi 32775	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) ↔ {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
5	3, 4	sylib 208	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   = wceq 1483  Ⅎwnf 1708  {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

This theorem is referenced by:  bj-abbidv  32779