Theorem abbid 2833

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 2179 and ax-11 2195. (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 2252	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		abbi 2830	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
5	3, 4	syl 18	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  Ⅎwnf 1816  {cab 2743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757

This theorem is used by:  rabbida4  3443  sbcbid  3800  sbceqbidf  32904  opabdm  33027  opabrn  33028  fpwrelmap  33148  sticksstones16  42987  rabbida2  45908  rabbida3  45911