Theorem oprabbid 5563

Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
oprabbid.1	|- F/xph
oprabbid.2	|- F/yph
oprabbid.3	|- F/zph
oprabbid.4	|- (ph -> (ps <-> ch))

Assertion

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

Distinct variable groups:   x,z   y,z

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

Proof of Theorem oprabbid

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oprabbid.1	. . . 4 |- F/xph
2		oprabbid.2	. . . . 5 |- F/yph
3		oprabbid.3	. . . . . 6 |- F/zph
4		oprabbid.4	. . . . . . 7 |- (ph -> (ps <-> ch))
5	4	anbi2d 684	. . . . . 6 |- (ph -> ((w = <.<.x, y>., z>. /\ ps) <-> (w = <.<.x, y>., z>. /\ ch)))
6	3, 5	exbid 1773	. . . . 5 |- (ph -> (E.z(w = <.<.x, y>., z>. /\ ps) <-> E.z(w = <.<.x, y>., z>. /\ ch)))
7	2, 6	exbid 1773	. . . 4 |- (ph -> (E.yE.z(w = <.<.x, y>., z>. /\ ps) <-> E.yE.z(w = <.<.x, y>., z>. /\ ch)))
8	1, 7	exbid 1773	. . 3 |- (ph -> (E.xE.yE.z(w = <.<.x, y>., z>. /\ ps) <-> E.xE.yE.z(w = <.<.x, y>., z>. /\ ch)))
9	8	abbidv 2467	. 2 |- (ph -> {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ps)} = {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ch)})
10		df-oprab 5528	. 2 |- {<.<.x, y>., z>. | ps} = {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ps)}
11		df-oprab 5528	. 2 |- {<.<.x, y>., z>. | ch} = {w | E.xE.yE.z(w = <.<.x, y>., z>. /\ ch)}
12	9, 10, 11	3eqtr4g 2410	1 |- (ph -> {<.<.x, y>., z>. | ps} = {<.<.x, y>., z>. | ch})

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642  {cab 2339  <.cop 4561  {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:  oprabbidv  5564  mpt2eq123  5661