Theorem submultl-P5 649

Description: Left Substitution Law for '⋅'.

Hypothesis

Ref	Expression
submultl-P5.1	⊢ (𝛾 → 𝑡 = 𝑢)

Assertion

Ref	Expression
submultl-P5	⊢ (𝛾 → (𝑡 ⋅ 𝑤) = (𝑢 ⋅ 𝑤))

Proof of Theorem submultl-P5

Step	Hyp	Ref	Expression
1		submultl-P5.1	. 2 ⊢ (𝛾 → 𝑡 = 𝑢)
2		ax-L9-multl 25	. 2 ⊢ (𝑡 = 𝑢 → (𝑡 ⋅ 𝑤) = (𝑢 ⋅ 𝑤))
3	1, 2	syl-P3.24.RC 260	1 ⊢ (𝛾 → (𝑡 ⋅ 𝑤) = (𝑢 ⋅ 𝑤))

Syntax hints:   ⋅ term-mult 5   = wff-equals 6   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-L9-multl 25

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  submultd-P5  651  ndsubmultl-P7.24d  854