Theorem abbid 2154

Description: Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-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 1415	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		abbi 2151	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) ↔ {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
5	3, 4	sylib 127	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  Ⅎwnf 1349  {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033

This theorem is referenced by:  abbidv  2155  rabeqf  2550  sbcbid  2816