Theorem oprabbid 7422

Description: Equivalent wff's yield equal operation class abstractions (deduction form). (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 636	. . . . . 6 ⊢ (𝜑 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
6	3, 5	exbid 2235	. . . . 5 ⊢ (𝜑 → (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
7	2, 6	exbid 2235	. . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
8	1, 7	exbid 2235	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
9	8	abbidv 2805	. 2 ⊢ (𝜑 → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)})
10		df-oprab 7361	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
11		df-oprab 7361	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4g 2799	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786  Ⅎwnf 1790  {cab 2717  ⟨cop 4562  {coprab 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-oprab 7361

This theorem is referenced by:  mpoeq123  7429