Theorem oprabbii 5566

Description: Equivalent wff's yield equal operation class abstractions. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors, 28-May-1995.) (Revised by set.mm contributors, 24-Jul-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 2353	. 2 ⊢ w = w
2		oprabbii.1	. . . 4 ⊢ (φ ↔ ψ)
3	2	a1i 10	. . 3 ⊢ (w = w → (φ ↔ ψ))
4	3	oprabbidv 5565	. 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}

This proof depends on syntax axioms:   ↔ wb 176   = wceq 1642  {coprab 5528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-oprab 5529

This theorem is used by:  oprab4  5567  oprabbi2i  5648  mpt2v  5720