Theorem oprabbidv 5611

Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.)

Hypothesis

Ref	Expression
oprabbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
oprabbidv	⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

Distinct variable groups:   𝑥,𝑧,𝜑   𝑦,𝑧,𝜑

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem oprabbidv

Step	Hyp	Ref	Expression
1		nfv 1462	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1462	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 1462	. 2 ⊢ Ⅎ𝑧𝜑
4		oprabbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	1, 2, 3, 4	oprabbid 5610	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1285  {coprab 5565

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-oprab 5568

This theorem is referenced by:  oprabbii  5612  mpt2eq123dva  5618  mpt2eq3dva  5621  resoprab2  5650  erovlem  6286