Theorem abbid 2467

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 1765	. 2 ⊢ (φ → ∀x(ψ ↔ χ))
4		abbib 2464	. 2 ⊢ ({x ∣ ψ} = {x ∣ χ} ↔ ∀x(ψ ↔ χ))
5	3, 4	sylibr 203	1 ⊢ (φ → {x ∣ ψ} = {x ∣ χ})

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346

This theorem is referenced by:  abbidv  2468  rabeqf  2853  sbcbid  3100