Theorem subeqr-P5-L1 634

Description: Lemma for subeqr-P5 635.

Assertion

Ref	Expression
subeqr-P5-L1	⊢ (𝑡 = 𝑢 → (𝑤 = 𝑡 → 𝑤 = 𝑢))

Proof of Theorem subeqr-P5-L1

Step	Hyp	Ref	Expression
1		ax-L7 19	. 2 ⊢ (𝑡 = 𝑢 → (𝑡 = 𝑤 → 𝑢 = 𝑤))
2		eqsym-P5.CL.SYM 629	. . 3 ⊢ (𝑡 = 𝑤 ↔ 𝑤 = 𝑡)
3		eqsym-P5.CL.SYM 629	. . 3 ⊢ (𝑢 = 𝑤 ↔ 𝑤 = 𝑢)
4	2, 3	subimd-P3.40c.RC 330	. 2 ⊢ ((𝑡 = 𝑤 → 𝑢 = 𝑤) ↔ (𝑤 = 𝑡 → 𝑤 = 𝑢))
5	1, 4	subimr2-P4.RC 543	1 ⊢ (𝑡 = 𝑢 → (𝑤 = 𝑡 → 𝑤 = 𝑢))

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:  subeqr-P5  635