Theorem cbvriotav 5749

Description: Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
cbvriotav.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvriotav	|- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Distinct variable groups:   x, A   y, A   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvriotav

Step	Hyp	Ref	Expression
1		nfv 1509	. 2 |- F/ y ph
2		nfv 1509	. 2 |- F/ x ps
3		cbvriotav.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvriota 5748	1 |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   iota_crio 5737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-sn 3538  df-uni 3745  df-iota 5096  df-riota 5738

This theorem is referenced by:  axcaucvg  7732