Theorem preq2 3439

Description: Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
preq2	⊢ (A = B → {𝐶, A} = {𝐶, B})

Proof of Theorem preq2

Step	Hyp	Ref	Expression
1		preq1 3438	. 2 ⊢ (A = B → {A, 𝐶} = {B, 𝐶})
2		prcom 3437	. 2 ⊢ {𝐶, A} = {A, 𝐶}
3		prcom 3437	. 2 ⊢ {𝐶, B} = {B, 𝐶}
4	1, 2, 3	3eqtr4g 2094	1 ⊢ (A = B → {𝐶, A} = {𝐶, B})

Syntax hints:   → wi 4   = wceq 1242  {cpr 3368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by:  preq12  3440  preq2i  3442  preq2d  3445  tpeq2  3448  preq12bg  3535  opeq2  3541  uniprg  3586  intprg  3639  prexgOLD  3937  prexg  3938  opth  3965  opeqsn  3980  relop  4429  funopg  4877  bj-prexg  9342