Theorem oprabbid 5817

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 459	. . . . . 6 ⊢ (𝜑 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
6	3, 5	exbid 1595	. . . . 5 ⊢ (𝜑 → (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
7	2, 6	exbid 1595	. . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
8	1, 7	exbid 1595	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
9	8	abbidv 2255	. 2 ⊢ (𝜑 → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)})
10		df-oprab 5771	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
11		df-oprab 5771	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)}
12	9, 10, 11	3eqtr4g 2195	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  Ⅎwnf 1436  ∃wex 1468  {cab 2123  ⟨cop 3525  {coprab 5768

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-oprab 5771

This theorem is referenced by:  oprabbidv  5818  mpoeq123  5823