MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixpeq2dv Structured version   Visualization version   GIF version

Theorem ixpeq2dv 8854
Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
ixpeq2dv.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
ixpeq2dv (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpeq2dv
StepHypRef Expression
1 ixpeq2dv.1 . . 3 (𝜑𝐵 = 𝐶)
21adantr 482 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
32ixpeq2dva 8853 1 (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Xcixp 8838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3446  df-in 3918  df-ss 3928  df-ixp 8839
This theorem is referenced by:  prdsval  17342  brssc  17702  isfunc  17755  natfval  17838  isnat  17839  dprdval  19787  elpt  22939  elptr  22940  dfac14  22985  hoicvrrex  44883  ovncvrrp  44891  ovnsubaddlem1  44897  ovnsubadd  44899  hoidmvlelem3  44924  hoidmvle  44927  ovnhoilem1  44928  ovnhoilem2  44929  ovnhoi  44930  hspval  44936  ovncvr2  44938  hspmbllem2  44954  hspmbl  44956  hoimbl  44958  opnvonmbl  44961  ovnovollem1  44983  ovnovollem3  44985  iinhoiicclem  45000  iinhoiicc  45001  vonioolem2  45008  vonioo  45009  vonicclem2  45011  vonicc  45012
  Copyright terms: Public domain W3C validator