Theorem oprabbii 5778

Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypothesis

Ref	Expression
oprabbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
oprabbii	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem oprabbii

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2113	. 2 ⊢ 𝑤 = 𝑤
2		oprabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓))
4	3	oprabbidv 5777	. 2 ⊢ (𝑤 = 𝑤 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
5	1, 4	ax-mp 7	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   ↔ wb 104   = wceq 1312  {coprab 5727

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-11 1465  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-oprab 5730

This theorem is referenced by:  oprab4  5794  mpov  5813  dfxp3  6044  tposmpo  6130  oviec  6487  dfplpq2  7104  dfmpq2  7105  dfmq0qs  7179  dfplq0qs  7180  addsrpr  7482  mulsrpr  7483  addcnsr  7563  mulcnsr  7564  addvalex  7573