Theorem opeq12 4580

Description: Equality theorem for ordered pairs. (Contributed by SF, 2-Jan-2015.)

Assertion

Ref	Expression
opeq12	⊢ ((A = B ∧ C = D) → ⟨A, C⟩ = ⟨B, D⟩)

Proof of Theorem opeq12

Step	Hyp	Ref	Expression
1		opeq1 4578	. 2 ⊢ (A = B → ⟨A, C⟩ = ⟨B, C⟩)
2		opeq2 4579	. 2 ⊢ (C = D → ⟨B, C⟩ = ⟨B, D⟩)
3	1, 2	sylan9eq 2405	1 ⊢ ((A = B ∧ C = D) → ⟨A, C⟩ = ⟨B, D⟩)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ⟨cop 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566

This theorem is referenced by:  opth  4602  copsex2t  4608  copsex2g  4609  cbvopab  4630  fsn  5432  fnressn  5438  cbvoprab12  5569