Theorem biopabd 2101

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

Hypotheses

Ref	Expression
biopabd.1	⊢ (φ → ∀xφ)
biopabd.2	⊢ (φ → ∀yφ)
biopabd.3	⊢ (φ → (ψ ↔ χ))

Assertion

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

Proof of Theorem biopabd

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ (φ → ∀zφ)
2		biopabd.1	. . . 4 ⊢ (φ → ∀xφ)
3		biopabd.2	. . . . 5 ⊢ (φ → ∀yφ)
4		biopabd.3	. . . . . 6 ⊢ (φ → (ψ ↔ χ))
5	4	anbi2d 468	. . . . 5 ⊢ (φ → ((z = ⟨x, y⟩ ∧ ψ) ↔ (z = ⟨x, y⟩ ∧ χ)))
6	3, 5	biexd 783	. . . 4 ⊢ (φ → (∃y(z = ⟨x, y⟩ ∧ ψ) ↔ ∃y(z = ⟨x, y⟩ ∧ χ)))
7	2, 6	biexd 783	. . 3 ⊢ (φ → (∃x∃y(z = ⟨x, y⟩ ∧ ψ) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ χ)))
8	1, 7	biabd 1182	. 2 ⊢ (φ → {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ χ)})
9		df-opab 2098	. 2 ⊢ {⟨x, y⟩∣ψ} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)}
10		df-opab 2098	. 2 ⊢ {⟨x, y⟩∣χ} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ χ)}
11	8, 9, 10	3eqtr4g 1147	1 ⊢ (φ → {⟨x, y⟩∣ψ} = {⟨x, y⟩∣χ})

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091  ⟨cop 1810  {copab 2055

This theorem is referenced by:  biopabdv 2102  bioprabd 3025  mapxpen 3390  xpmapenlem3 3393  xpmapenlem4 3394  xpmapenlem5 3395

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  df-opab 2098

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