Theorem oprabbidv 5896

Description: Equivalent wff's yield equal operation class abstractions (deduction form). (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 1516	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1516	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 1516	. 2 ⊢ Ⅎ𝑧𝜑
4		oprabbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	1, 2, 3, 4	oprabbid 5895	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  {coprab 5843

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-oprab 5846

This theorem is referenced by:  oprabbii  5897  mpoeq123dva  5903  mpoeq3dva  5906  resoprab2  5939  erovlem  6593