Theorem biabd 1182

Description: Equivalent wff's yield equal class abstractions (deduction rule).

Hypotheses

Ref	Expression
biabd.1	⊢ (φ → ∀xφ)
biabd.2	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
biabd	⊢ (φ → {x∣ψ} = {x∣χ})

Proof of Theorem biabd

Step	Hyp	Ref	Expression
1		biabd.1	. . 3 ⊢ (φ → ∀xφ)
2		biabd.2	. . 3 ⊢ (φ → (ψ ↔ χ))
3	1, 2	19.21ai 740	. 2 ⊢ (φ → ∀x(ψ ↔ χ))
4		cleq2ab 1179	. 2 ⊢ ({x∣ψ} = {x∣χ} ↔ ∀x(ψ ↔ χ))
5	3, 4	sylibr 175	1 ⊢ (φ → {x∣ψ} = {x∣χ})

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  {cab 1090   = wceq 1091

This theorem is referenced by:  biabdv 1183  rabeqf 1345  moabex 1868  biopabd 2101

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

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