Theorem eqtrns-P5 630

Description: Equivalence Property: '=' Transitivity.

Hypotheses

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

Assertion

Ref	Expression
eqtrns-P5	⊢ (𝛾 → 𝑡 = 𝑤)

Proof of Theorem eqtrns-P5

Step	Hyp	Ref	Expression
1		eqtrns-P5.2	. 2 ⊢ (𝛾 → 𝑢 = 𝑤)
2		eqtrns-P5.1	. . 3 ⊢ (𝛾 → 𝑡 = 𝑢)
3		ax-L7 19	. . . . 5 ⊢ (𝑢 = 𝑡 → (𝑢 = 𝑤 → 𝑡 = 𝑤))
4		eqsym-P5.CL.SYM 629	. . . . 5 ⊢ (𝑢 = 𝑡 ↔ 𝑡 = 𝑢)
5	3, 4	subiml2-P4.RC 541	. . . 4 ⊢ (𝑡 = 𝑢 → (𝑢 = 𝑤 → 𝑡 = 𝑤))
6	5	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → (𝑡 = 𝑢 → (𝑢 = 𝑤 → 𝑡 = 𝑤)))
7	2, 6	ndime-P3.6 171	. 2 ⊢ (𝛾 → (𝑢 = 𝑤 → 𝑡 = 𝑤))
8	1, 7	ndime-P3.6 171	1 ⊢ (𝛾 → 𝑡 = 𝑤)

Syntax hints:   = 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

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:  eqtrns-P5.CL  631  subaddd-P5  647  submultd-P5  651  spliteq-P6-L1  775  psubsuccv-P6-L1  805  psubaddv-P6-L1  807  psubmultv-P6-L1  809