Theorem ixpeq2dv 8952

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 8951	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Xcixp 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-ss 3980  df-ixp 8937

This theorem is referenced by:  prdsval  17502  brssc  17862  isfunc  17915  natfval  18001  isnat  18002  dprdval  20038  elpt  23596  elptr  23597  dfac14  23642  ixpeq12dv  36199  hoicvrrex  46512  ovncvrrp  46520  ovnsubaddlem1  46526  ovnsubadd  46528  hoidmvlelem3  46553  hoidmvle  46556  ovnhoilem1  46557  ovnhoilem2  46558  ovnhoi  46559  hspval  46565  ovncvr2  46567  hspmbllem2  46583  hspmbl  46585  hoimbl  46587  opnvonmbl  46590  ovnovollem1  46612  ovnovollem3  46614  iinhoiicclem  46629  iinhoiicc  46630  vonioolem2  46637  vonioo  46638  vonicclem2  46640  vonicc  46641