Theorem subaddd-P5 647

Description: Dual Substitution Law for '+'.

Hypotheses

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

Assertion

Ref	Expression
subaddd-P5	⊢ (𝛾 → (𝑠 + 𝑢) = (𝑡 + 𝑤))

Proof of Theorem subaddd-P5

Step	Hyp	Ref	Expression
1		subaddd-P5.1	. . 3 ⊢ (𝛾 → 𝑠 = 𝑡)
2	1	subaddl-P5 645	. 2 ⊢ (𝛾 → (𝑠 + 𝑢) = (𝑡 + 𝑢))
3		subaddd-P5.2	. . 3 ⊢ (𝛾 → 𝑢 = 𝑤)
4	3	subaddr-P5 646	. 2 ⊢ (𝛾 → (𝑡 + 𝑢) = (𝑡 + 𝑤))
5	2, 4	eqtrns-P5 630	1 ⊢ (𝛾 → (𝑠 + 𝑢) = (𝑡 + 𝑤))

Syntax hints:   + term-add 4   = 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-addl 23  ax-L9-addr 24

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:  subaddd-P5.CL  648  psubaddv-P6-L1  807