Theorem tpeq123d 3799

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 3796	. 2 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸})
3		tpeq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	tpeq2d 3797	. 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸})
5		tpeq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	tpeq3d 3798	. 2 ⊢ (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹})
7	2, 4, 6	3eqtrd 2275	1 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})

Syntax hints:   → wi 4   = wceq 1402  {ctp 3707

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-tp 3713

This theorem is referenced by:  fz0tp  10507  fz0to4untppr  10509  fzo0to3tp  10615  imasex  13603  imasival  13604  prdsex  14149  prdsval  14150  psrval  14973