Theorem abbid 2346

Description: Equivalent wff's yield equal class abstractions (deduction form). (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 1568	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		abbi 2343	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) ↔ {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
5	3, 4	sylib 122	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393   = wceq 1395  Ⅎwnf 1506  {cab 2215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222

This theorem is referenced by:  abbidv  2347  rabeqf  2789  sbcbid  3086