Theorem cbvriotav 6051

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 1581	. 2 |- F/ y ph
2		nfv 1581	. 2 |- F/ x ps
3		cbvriotav.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvriota 6050	1 |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   iota_crio 6037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-sn 3715  df-uni 3936  df-iota 5337  df-riota 6038

This theorem is used by:  axcaucvg  8267