Theorem oprabbii 7423

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 2734	. 2 ⊢ 𝑤 = 𝑤
2		oprabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 11	. . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓))
4	3	oprabbidv 7422	. 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 206   = wceq 1541  {coprab 7357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-oprab 7360

This theorem is referenced by:  oprab4  7442  mpov  7468  dfxp3  8003  tposmpo  8203  addsrpr  10984  mulsrpr  10985  addcnsr  11044  mulcnsr  11045  joinfval2  18293  meetfval2  18307  dfxrn2  38509