Theorem opabbid 4715

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 740	. . . . 5 ⊢ (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
5	2, 4	exbid 2091	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
6	1, 5	exbid 2091	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
7	6	abbidv 2741	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)})
8		df-opab 4713	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
9		df-opab 4713	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)}
10	7, 8, 9	3eqtr4g 2681	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704  Ⅎwnf 1708  {cab 2608  ⟨cop 4183  {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-opab 4713

This theorem is referenced by:  opabbidv  4716  mpteq12f  4731  mpteq12d  4734  mpteq12df  4735  feqmptdf  6251  fnoprabg  6761  sprsymrelfo  41747