Theorem ixpeq2dv 8903

Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.)

Hypothesis

Ref	Expression
ixpeq2dv.1	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
ixpeq2dv	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem ixpeq2dv

Step	Hyp	Ref	Expression
1		ixpeq2dv.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
2	1	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	ixpeq2dva 8902	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Xcixp 8887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-v 3476  df-in 3954  df-ss 3964  df-ixp 8888

This theorem is referenced by:  prdsval  17397  brssc  17757  isfunc  17810  natfval  17893  isnat  17894  dprdval  19867  elpt  23067  elptr  23068  dfac14  23113  hoicvrrex  45258  ovncvrrp  45266  ovnsubaddlem1  45272  ovnsubadd  45274  hoidmvlelem3  45299  hoidmvle  45302  ovnhoilem1  45303  ovnhoilem2  45304  ovnhoi  45305  hspval  45311  ovncvr2  45313  hspmbllem2  45329  hspmbl  45331  hoimbl  45333  opnvonmbl  45336  ovnovollem1  45358  ovnovollem3  45360  iinhoiicclem  45375  iinhoiicc  45376  vonioolem2  45383  vonioo  45384  vonicclem2  45386  vonicc  45387