Theorem opabbid 4625

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Hypotheses

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

Assertion

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

Proof of Theorem opabbid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabbid.1	. . . 4 ⊢ Ⅎxφ
2		opabbid.2	. . . . 5 ⊢ Ⅎyφ
3		opabbid.3	. . . . . 6 ⊢ (φ → (ψ ↔ χ))
4	3	anbi2d 684	. . . . 5 ⊢ (φ → ((z = ⟨x, y⟩ ∧ ψ) ↔ (z = ⟨x, y⟩ ∧ χ)))
5	2, 4	exbid 1773	. . . 4 ⊢ (φ → (∃y(z = ⟨x, y⟩ ∧ ψ) ↔ ∃y(z = ⟨x, y⟩ ∧ χ)))
6	1, 5	exbid 1773	. . 3 ⊢ (φ → (∃x∃y(z = ⟨x, y⟩ ∧ ψ) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ χ)))
7	6	abbidv 2468	. 2 ⊢ (φ → {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ ψ)} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ χ)})
8		df-opab 4624	. 2 ⊢ {⟨x, y⟩ ∣ ψ} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ ψ)}
9		df-opab 4624	. 2 ⊢ {⟨x, y⟩ ∣ χ} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ χ)}
10	7, 8, 9	3eqtr4g 2410	1 ⊢ (φ → {⟨x, y⟩ ∣ ψ} = {⟨x, y⟩ ∣ χ})

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  ⟨cop 4562  {copab 4623

This proof depends on 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 proof 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-opab 4624

This theorem is used by:  opabbidv  4626  fnoprabg  5586  mpteq12f  5656