Theorem oprabbii 6945

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 2800	. 2 ⊢ 𝑤 = 𝑤
2		oprabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 11	. . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓))
4	3	oprabbidv 6944	. 2 ⊢ (𝑤 = 𝑤 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
5	1, 4	ax-mp 5	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   ↔ wb 198   = wceq 1653  {coprab 6880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-oprab 6883

This theorem is referenced by:  oprab4  6961  mpt2v  6985  dfxp3  7467  tposmpt2  7628  addsrpr  10185  mulsrpr  10186  addcnsr  10245  mulcnsr  10246  joinfval2  17316  meetfval2  17330  dfxrn2  34631