Theorem oprabbii 5565

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	|- (ph <-> ps)

Assertion

Ref	Expression
oprabbii	|- {<.<.x, y>., z>. | ph} = {<.<.x, y>., z>. | ps}

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   ph(x,y,z)   ps(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 |- (ph <-> ps)
3	2	a1i 10	. . 3 |- (w = w -> (ph <-> ps))
4	3	oprabbidv 5564	. 2 |- (w = w -> {<.<.x, y>., z>. | ph} = {<.<.x, y>., z>. | ps})
5	1, 4	ax-mp 5	1 |- {<.<.x, y>., z>. | ph} = {<.<.x, y>., z>. | ps}

Syntax hints:   <-> wb 176   = wceq 1642  {coprab 5527

This theorem was proved from 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 theorem 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 5528

This theorem is referenced by:  oprab4  5566  oprabbi2i  5647  mpt2v  5719