Theorem bioprabdv 3026

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

Hypothesis

Ref	Expression
bioprabdv.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
bioprabdv	⊢ (φ → {⟨⟨x, y⟩, z⟩∣ψ} = {⟨⟨x, y⟩, z⟩∣χ})

Distinct variable group(s):   x,z,φ   y,z,φ

Proof of Theorem bioprabdv

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (φ → ∀xφ)
2		ax-17 925	. 2 ⊢ (φ → ∀yφ)
3		ax-17 925	. 2 ⊢ (φ → ∀zφ)
4		bioprabdv.1	. 2 ⊢ (φ → (ψ ↔ χ))
5	1, 2, 3, 4	bioprabd 3025	1 ⊢ (φ → {⟨⟨x, y⟩, z⟩∣ψ} = {⟨⟨x, y⟩, z⟩∣χ})

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091  {copab2 3002

This theorem is referenced by:  bioprabi 3027

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-oprab 3004

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