Theorem submultd-P5.CL 652

Description: Closed Form of submultd-P5 651.

Assertion

Ref	Expression
submultd-P5.CL	⊢ ((𝑠 = 𝑡 ∧ 𝑢 = 𝑤) → (𝑠 ⋅ 𝑢) = (𝑡 ⋅ 𝑤))

Proof of Theorem submultd-P5.CL

Step	Hyp	Ref	Expression
1		rcp-NDASM1of2 193	. 2 ⊢ ((𝑠 = 𝑡 ∧ 𝑢 = 𝑤) → 𝑠 = 𝑡)
2		rcp-NDASM2of2 194	. 2 ⊢ ((𝑠 = 𝑡 ∧ 𝑢 = 𝑤) → 𝑢 = 𝑤)
3	1, 2	submultd-P5 651	1 ⊢ ((𝑠 = 𝑡 ∧ 𝑢 = 𝑤) → (𝑠 ⋅ 𝑢) = (𝑡 ⋅ 𝑤))

Syntax hints:   ⋅ term-mult 5   = wff-equals 6   → wff-imp 10   ∧ wff-and 132

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:  nfrmult-P6  782