Theorem abbid 2810

Description: Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) Avoid ax-10 2139 and ax-11 2156. (Revised by Wolf Lammen, 6-May-2023.)

Hypotheses

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

Assertion

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

Proof of Theorem abbid

Step	Hyp	Ref	Expression
1		abbid.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		abbid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	alrimi 2209	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		abbi1 2807	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
5	3, 4	syl 17	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  Ⅎwnf 1787  {cab 2715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730

This theorem is referenced by:  rabeqf  3405  rabeqiOLD  3407  sbcbid  3769  sbceqbidf  30736  opabdm  30852  opabrn  30853  fpwrelmap  30970  bj-rabbida2  35033  sticksstones16  40046  rabbida2  42570  rabbida3  42573