Theorem submultd-P5 651

Description: Dual Substitution Law for '⋅'.

Hypotheses

Ref	Expression
submultd-P5.1	⊢ (𝛾 → 𝑠 = 𝑡)
submultd-P5.2	⊢ (𝛾 → 𝑢 = 𝑤)

Assertion

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

Proof of Theorem submultd-P5

Step	Hyp	Ref	Expression
1		submultd-P5.1	. . 3 ⊢ (𝛾 → 𝑠 = 𝑡)
2	1	submultl-P5 649	. 2 ⊢ (𝛾 → (𝑠 ⋅ 𝑢) = (𝑡 ⋅ 𝑢))
3		submultd-P5.2	. . 3 ⊢ (𝛾 → 𝑢 = 𝑤)
4	3	submultr-P5 650	. 2 ⊢ (𝛾 → (𝑡 ⋅ 𝑢) = (𝑡 ⋅ 𝑤))
5	2, 4	eqtrns-P5 630	1 ⊢ (𝛾 → (𝑠 ⋅ 𝑢) = (𝑡 ⋅ 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-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L9-multl 25  ax-L9-multr 26

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596

This theorem is referenced by:  submultd-P5.CL  652  psubmultv-P6-L1  809