Theorem xpeq1 5638

Description: Equality theorem for Cartesian product. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
xpeq1	⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Proof of Theorem xpeq1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2826	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	anbi1d 632	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
3	2	opabbidv 5152	. 2 ⊢ (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)})
4		df-xp 5630	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)}
5		df-xp 5630	. 2 ⊢ (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
6	3, 4, 5	3eqtr4g 2797	1 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {copab 5148   × cxp 5622

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-opab 5149  df-xp 5630

This theorem is referenced by:  xpeq12  5649  xpeq1i  5650  xpeq1d  5653  opthprc  5688  dmxpid  5879  reseq2  5933  xpnz  6117  xpdisj1  6119  xpcan2  6135  xpima  6140  unixp  6240  unixpid  6242  naddcllem  8605  pmvalg  8777  xpsneng  8993  xpcomeng  9000  xpdom2g  9004  fodomr  9059  unxpdom  9162  fodomfir  9231  marypha1lem  9339  iundom2g  10453  hashxplem  14386  dmtrclfv  14971  ramcl  16991  efgval  19683  frgpval  19724  frlmval  21738  txuni2  23540  txbas  23542  txopn  23577  txrest  23606  txdis  23607  txdis1cn  23610  tx1stc  23625  tmdgsum  24070  qustgplem  24096  incistruhgr  29162  isgrpo  30583  hhssablo  31349  hhssnvt  31351  hhsssh  31355  gsumpart  33139  txomap  33994  tpr2rico  34072  elsx  34354  br2base  34429  dya2iocnrect  34441  sxbrsigalem5  34448  sibf0  34494  cvmlift2lem13  35513