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Theorem abbid 2885
 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 2154. (Revised by Wolf Lammen, 6-May-2023.)
Hypotheses
Ref Expression
abbid.1 𝑥𝜑
abbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
abbid (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Proof of Theorem abbid
StepHypRef Expression
1 abbid.1 . . 3 𝑥𝜑
2 abbid.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimi 2206 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 abbi1 2882 . 2 (∀𝑥(𝜓𝜒) → {𝑥𝜓} = {𝑥𝜒})
53, 4syl 17 1 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1529   = wceq 1531  Ⅎwnf 1778  {cab 2797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-9 2118  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812 This theorem is referenced by:  rabeqf  3480  rabeqi  3481  sbcbid  3824  sbceqbidf  30242  opabdm  30354  opabrn  30355  fpwrelmap  30461  bj-rabbida2  34230  rabbida2  41388  rabbida3  41391
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