Theorem ixpeq2dv 8863

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 8862	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Xcixp 8847

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-ss 3920  df-ixp 8848

This theorem is referenced by:  prdsval  17387  brssc  17750  isfunc  17800  natfval  17885  isnat  17886  dprdval  19946  elpt  23528  elptr  23529  dfac14  23574  ixpeq12dv  36432  hoicvrrex  46914  ovncvrrp  46922  ovnsubaddlem1  46928  ovnsubadd  46930  hoidmvlelem3  46955  hoidmvle  46958  ovnhoilem1  46959  ovnhoilem2  46960  ovnhoi  46961  hspval  46967  ovncvr2  46969  hspmbllem2  46985  hspmbl  46987  hoimbl  46989  opnvonmbl  46992  ovnovollem1  47014  ovnovollem3  47016  iinhoiicclem  47031  iinhoiicc  47032  vonioolem2  47039  vonioo  47040  vonicclem2  47042  vonicc  47043