Theorem eqnetrrd 2225

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

Hypotheses

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

Assertion

Ref	Expression
eqnetrrd	⊢ (φ → B ≠ 𝐶)

Proof of Theorem eqnetrrd

Step	Hyp	Ref	Expression
1		eqnetrrd.1	. . 3 ⊢ (φ → A = B)
2	1	eqcomd 2042	. 2 ⊢ (φ → B = A)
3		eqnetrrd.2	. 2 ⊢ (φ → A ≠ 𝐶)
4	2, 3	eqnetrd 2223	1 ⊢ (φ → B ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1242   ≠ wne 2201

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-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-ne 2203

This theorem is referenced by: (None)