Theorem ixpeq2dv 8861

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 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	ixpeq2dva 8860	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Xcixp 8845

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-ss 3906  df-ixp 8846

This theorem is referenced by:  prdsval  17418  brssc  17781  isfunc  17831  natfval  17916  isnat  17917  dprdval  19980  elpt  23537  elptr  23538  dfac14  23583  ixpeq12dv  36398  hoicvrrex  46984  ovncvrrp  46992  ovnsubaddlem1  46998  ovnsubadd  47000  hoidmvlelem3  47025  hoidmvle  47028  ovnhoilem1  47029  ovnhoilem2  47030  ovnhoi  47031  hspval  47037  ovncvr2  47039  hspmbllem2  47055  hspmbl  47057  hoimbl  47059  opnvonmbl  47062  ovnovollem1  47084  ovnovollem3  47086  iinhoiicclem  47101  iinhoiicc  47102  vonioolem2  47109  vonioo  47110  vonicclem2  47112  vonicc  47113