Theorem cbvriotav 5820

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 1521	. 2 |- F/ y ph
2		nfv 1521	. 2 |- F/ x ps
3		cbvriotav.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvriota 5819	1 |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   iota_crio 5808

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-sn 3589  df-uni 3797  df-iota 5160  df-riota 5809

This theorem is referenced by:  axcaucvg  7862