Theorem cbvixpv 8927

Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
cbvixpv.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvixpv	⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvixpv

Step	Hyp	Ref	Expression
1		nfcv 2899	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2899	. 2 ⊢ Ⅎ𝑥𝐶
3		cbvixpv.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbvixp 8926	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1534  Xcixp 8909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fn 6545  df-fv 6550  df-ixp 8910

This theorem is referenced by:  funcpropd  17882  invfuc  17959  natpropd  17961  dprdw  19960  dprdwd  19961  ptuni2  23473  ptbasin  23474  ptbasfi  23478  ptpjopn  23509  ptclsg  23512  dfac14  23515  ptcnp  23519  ptcmplem2  23950  ptcmpg  23954  prdsxmslem2  24431  upixp  37196  rrxsnicc  45682  ioorrnopn  45687  ioorrnopnxr  45689  ovnsubadd  45954  hoidmvlelem4  45980  hoidmvle  45982  hspdifhsp  45998  hoiqssbllem2  46005  hspmbl  46011  hoimbl  46013  opnvonmbl  46016  ovnovollem3  46040