Theorem opabbid 5167

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (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 639	. . . . 5 ⊢ (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
5	2, 4	exbid 2260	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
6	1, 5	exbid 2260	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
7	6	abbidv 2830	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)})
8		df-opab 5165	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
9		df-opab 5165	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)}
10	7, 8, 9	3eqtr4g 2824	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  ∃wex 1801  Ⅎwnf 1805  {cab 2742  ⟨cop 4590  {copab 5164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-9 2154  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-opab 5165

This theorem is referenced by:  mpteq12da  5185  mpteq12f  5187  feqmptdf  6939  fnoprabg  7521  sprsymrelfo  48108