Theorem oprabbii 7500

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 2737	. 2 ⊢ 𝑤 = 𝑤
2		oprabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 11	. . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓))
4	3	oprabbidv 7499	. 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 1540  {coprab 7432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-oprab 7435

This theorem is referenced by:  oprab4  7519  mpov  7545  dfxp3  8086  tposmpo  8288  addsrpr  11115  mulsrpr  11116  addcnsr  11175  mulcnsr  11176  joinfval2  18419  meetfval2  18433  dfxrn2  38377