Theorem tpeq123d 3763

Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Hypotheses

Ref	Expression
tpeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
tpeq123d.2	⊢ (𝜑 → 𝐶 = 𝐷)
tpeq123d.3	⊢ (𝜑 → 𝐸 = 𝐹)

Assertion

Ref	Expression
tpeq123d	⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Proof of Theorem tpeq123d

Step	Hyp	Ref	Expression
1		tpeq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	tpeq1d 3760	. 2 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3		tpeq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	tpeq2d 3761	. 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5		tpeq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	tpeq3d 3762	. 2 ⊢ (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
7	2, 4, 6	3eqtrd 2268	1 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Syntax hints:   → wi 4   = wceq 1397  {ctp 3671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-tp 3677

This theorem is referenced by:  fz0tp  10356  fz0to4untppr  10358  fzo0to3tp  10463  prdsex  13351  prdsval  13355  imasex  13387  imasival  13388  psrval  14679