Theorem cbvriotav 5844

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 1528	. 2 |- F/ y ph
2		nfv 1528	. 2 |- F/ x ps
3		cbvriotav.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvriota 5843	1 |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   iota_crio 5832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-sn 3600  df-uni 3812  df-iota 5180  df-riota 5833

This theorem is referenced by:  axcaucvg  7901