Theorem oprabbid 5563

Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
oprabbid.1	⊢ Ⅎxφ
oprabbid.2	⊢ Ⅎyφ
oprabbid.3	⊢ Ⅎzφ
oprabbid.4	⊢ (φ → (ψ ↔ χ))

Assertion

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

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)   χ(x,y,z)

Proof of Theorem oprabbid

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oprabbid.1	. . . 4 ⊢ Ⅎxφ
2		oprabbid.2	. . . . 5 ⊢ Ⅎyφ
3		oprabbid.3	. . . . . 6 ⊢ Ⅎzφ
4		oprabbid.4	. . . . . . 7 ⊢ (φ → (ψ ↔ χ))
5	4	anbi2d 684	. . . . . 6 ⊢ (φ → ((w = ⟨⟨x, y⟩, z⟩ ∧ ψ) ↔ (w = ⟨⟨x, y⟩, z⟩ ∧ χ)))
6	3, 5	exbid 1773	. . . . 5 ⊢ (φ → (∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ψ) ↔ ∃z(w = ⟨⟨x, y⟩, z⟩ ∧ χ)))
7	2, 6	exbid 1773	. . . 4 ⊢ (φ → (∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ψ) ↔ ∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ χ)))
8	1, 7	exbid 1773	. . 3 ⊢ (φ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ψ) ↔ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ χ)))
9	8	abbidv 2467	. 2 ⊢ (φ → {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ψ)} = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ χ)})
10		df-oprab 5528	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ψ} = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ψ)}
11		df-oprab 5528	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ χ} = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ χ)}
12	9, 10, 11	3eqtr4g 2410	1 ⊢ (φ → {⟨⟨x, y⟩, z⟩ ∣ ψ} = {⟨⟨x, y⟩, z⟩ ∣ χ})

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  ⟨cop 4561  {coprab 5527

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  df-oprab 5528

This theorem is referenced by:  oprabbidv  5564  mpt2eq123  5661