Theorem eqnetrd 2535

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Hypotheses

Ref	Expression
eqnetrd.1	⊢ (φ → A = B)
eqnetrd.2	⊢ (φ → B ≠ C)

Assertion

Ref	Expression
eqnetrd	⊢ (φ → A ≠ C)

Proof of Theorem eqnetrd

Step	Hyp	Ref	Expression
1		eqnetrd.2	. 2 ⊢ (φ → B ≠ C)
2		eqnetrd.1	. . 3 ⊢ (φ → A = B)
3	2	neeq1d 2530	. 2 ⊢ (φ → (A ≠ C ↔ B ≠ C))
4	1, 3	mpbird 223	1 ⊢ (φ → A ≠ C)

This proof depends on syntax axioms:   → wi 4   = wceq 1642   ≠ wne 2517

This proof depends on 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-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is used by:  eqnetrrd  2537  vfin1cltv  4548  nchoicelem12  6301  nchoicelem14  6303  nchoicelem17  6306