Theorem subeql-P5 632

Description: Left Substitution Law for '=' .

Hypothesis

Ref	Expression
subeql-P5.1	⊢ (𝛾 → 𝑡 = 𝑢)

Assertion

Ref	Expression
subeql-P5	⊢ (𝛾 → (𝑡 = 𝑤 ↔ 𝑢 = 𝑤))

Proof of Theorem subeql-P5

Step	Hyp	Ref	Expression
1		subeql-P5.1	. . 3 ⊢ (𝛾 → 𝑡 = 𝑢)
2		ax-L7 19	. . . 4 ⊢ (𝑡 = 𝑢 → (𝑡 = 𝑤 → 𝑢 = 𝑤))
3	2	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → (𝑡 = 𝑢 → (𝑡 = 𝑤 → 𝑢 = 𝑤)))
4	1, 3	ndime-P3.6 171	. 2 ⊢ (𝛾 → (𝑡 = 𝑤 → 𝑢 = 𝑤))
5	1	eqsym-P5 627	. . 3 ⊢ (𝛾 → 𝑢 = 𝑡)
6		ax-L7 19	. . . 4 ⊢ (𝑢 = 𝑡 → (𝑢 = 𝑤 → 𝑡 = 𝑤))
7	6	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → (𝑢 = 𝑡 → (𝑢 = 𝑤 → 𝑡 = 𝑤)))
8	5, 7	ndime-P3.6 171	. 2 ⊢ (𝛾 → (𝑢 = 𝑤 → 𝑡 = 𝑤))
9	4, 8	ndbii-P3.13 178	1 ⊢ (𝛾 → (𝑡 = 𝑤 ↔ 𝑢 = 𝑤))

Syntax hints:   = wff-equals 6   → wff-imp 10   ↔ wff-bi 104

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:  subeql-P5.CL  633  subeqd-P5  637  example-E5.02a  664  example-E5.04a  675  ndsubeql-P7.22a  847