Theorem ixpeq2dv 8333

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 8332	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  Xcixp 8317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-in 3872  df-ss 3880  df-ixp 8318

This theorem is referenced by:  prdsval  16561  brssc  16917  isfunc  16967  natfval  17049  isnat  17050  dprdval  18846  elpt  21868  elptr  21869  dfac14  21914  hoicvrrex  42402  ovncvrrp  42410  ovnsubaddlem1  42416  ovnsubadd  42418  hoidmvlelem3  42443  hoidmvle  42446  ovnhoilem1  42447  ovnhoilem2  42448  ovnhoi  42449  hspval  42455  ovncvr2  42457  hspmbllem2  42473  hspmbl  42475  hoimbl  42477  opnvonmbl  42480  ovnovollem1  42502  ovnovollem3  42504  iinhoiicclem  42519  iinhoiicc  42520  vonioolem2  42527  vonioo  42528  vonicclem2  42530  vonicc  42531