Theorem oprabbidv 6074

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 1576	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1576	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 1576	. 2 ⊢ Ⅎ𝑧𝜑
4		oprabbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	1, 2, 3, 4	oprabbid 6073	1 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  {coprab 6018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-oprab 6021

This theorem is referenced by:  oprabbii  6075  mpoeq123dva  6081  mpoeq3dva  6084  resoprab2  6117  erovlem  6795