Theorem tpeq123d 3815

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 3812	. 2 ⊢ (φ → {A, C, E} = {B, C, E})
3		tpeq123d.2	. . 3 ⊢ (φ → C = D)
4	3	tpeq2d 3813	. 2 ⊢ (φ → {B, C, E} = {B, D, E})
5		tpeq123d.3	. . 3 ⊢ (φ → E = F)
6	5	tpeq3d 3814	. 2 ⊢ (φ → {B, D, E} = {B, D, F})
7	2, 4, 6	3eqtrd 2389	1 ⊢ (φ → {A, C, E} = {B, D, F})

Syntax hints:   → wi 4   = wceq 1642  {ctp 3740

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-tp 3744

This theorem is referenced by: (None)