Theorem bioprabd 3025

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

Hypotheses

Ref	Expression
bioprabd.1	⊢ (φ → ∀xφ)
bioprabd.2	⊢ (φ → ∀yφ)
bioprabd.3	⊢ (φ → ∀zφ)
bioprabd.4	⊢ (φ → (ψ ↔ χ))

Assertion

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

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

Proof of Theorem bioprabd

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ (φ → ∀wφ)
2		bioprabd.3	. . 3 ⊢ (φ → ∀zφ)
3		bioprabd.1	. . . 4 ⊢ (φ → ∀xφ)
4		bioprabd.2	. . . . 5 ⊢ (φ → ∀yφ)
5		bioprabd.4	. . . . . 6 ⊢ (φ → (ψ ↔ χ))
6	5	anbi2d 468	. . . . 5 ⊢ (φ → ((w = ⟨x, y⟩ ∧ ψ) ↔ (w = ⟨x, y⟩ ∧ χ)))
7	4, 6	biexd 783	. . . 4 ⊢ (φ → (∃y(w = ⟨x, y⟩ ∧ ψ) ↔ ∃y(w = ⟨x, y⟩ ∧ χ)))
8	3, 7	biexd 783	. . 3 ⊢ (φ → (∃x∃y(w = ⟨x, y⟩ ∧ ψ) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ χ)))
9	1, 2, 8	biopabd 2101	. 2 ⊢ (φ → {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ ψ)} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ χ)})
10		dfoprab2 3021	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣ψ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ ψ)}
11		dfoprab2 3021	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣χ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ χ)}
12	9, 10, 11	3eqtr4g 1147	1 ⊢ (φ → {⟨⟨x, y⟩, z⟩∣ψ} = {⟨⟨x, y⟩, z⟩∣χ})

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091  ⟨cop 1810  {copab 2055  {copab2 3002

This theorem is referenced by:  bioprabdv 3026  mapxpen 3390

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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