Theorem abbid 2740

Description: Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

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 2082	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		abbi 2737	. 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-13 2246  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:  abbidv  2741  rabeqf  3190  sbcbid  3489  sbceqbidf  29321  opabdm  29423  opabrn  29424  fpwrelmap  29508  bj-rabbida2  32913  iotain  38618  rabbida2  39317  rabbida3  39320