Theorem opabbid 3993

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 459	. . . . 5 ⊢ (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
5	2, 4	exbid 1595	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
6	1, 5	exbid 1595	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
7	6	abbidv 2257	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)})
8		df-opab 3990	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
9		df-opab 3990	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)}
10	7, 8, 9	3eqtr4g 2197	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  Ⅎwnf 1436  ∃wex 1468  {cab 2125  ⟨cop 3530  {copab 3988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-opab 3990

This theorem is referenced by:  opabbidv  3994  mpteq12f  4008  fnoprabg  5872