Theorem oprabbidv 5565

Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.)

Hypothesis

Ref	Expression
oprabbidv.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
oprabbidv	|- (ph -> {<.<.x, y>., z>. | ps} = {<.<.x, y>., z>. | ch})

Distinct variable groups:   x,z,ph   y,z,ph

Allowed substitution hints:   ps(x,y,z)   ch(x,y,z)

Proof of Theorem oprabbidv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/xph
2		nfv 1619	. 2 |- F/yph
3		nfv 1619	. 2 |- F/zph
4		oprabbidv.1	. 2 |- (ph -> (ps <-> ch))
5	1, 2, 3, 4	oprabbid 5564	1 |- (ph -> {<.<.x, y>., z>. | ps} = {<.<.x, y>., z>. | ch})

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  {coprab 5528

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 5529

This theorem is referenced by:  oprabbii  5566  resoprab2  5583  mpt2eq123dv  5664  mpt2eq3dva  5670