Theorem syl5eqner 2236

Description: B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)

Hypotheses

Ref	Expression
syl5eqner.1	⊢ 𝐵 = 𝐴
syl5eqner.2	⊢ (𝜑 → 𝐵 ≠ 𝐶)

Assertion

Ref	Expression
syl5eqner	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem syl5eqner

Step	Hyp	Ref	Expression
1		syl5eqner.2	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
2		syl5eqner.1	. . 3 ⊢ 𝐵 = 𝐴
3	2	neeq1i 2220	. 2 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶)
4	1, 3	sylib 127	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1243   ≠ wne 2204

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 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-ne 2206

This theorem is referenced by: (None)