Theorem eqsym-P5 627

Description: Equivalence Property: '=' symmetry.

Hypothesis

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

Assertion

Ref	Expression
eqsym-P5	⊢ (𝛾 → 𝑢 = 𝑡)

Proof of Theorem eqsym-P5

Step	Hyp	Ref	Expression
1		eqsym-P5.1	. 2 ⊢ (𝛾 → 𝑡 = 𝑢)
2		ax-L7 19	. . . 4 ⊢ (𝑡 = 𝑢 → (𝑡 = 𝑡 → 𝑢 = 𝑡))
3	2	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → (𝑡 = 𝑢 → (𝑡 = 𝑡 → 𝑢 = 𝑡)))
4		eqref-P5 626	. . . 4 ⊢ 𝑡 = 𝑡
5	4	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → 𝑡 = 𝑡)
6	3, 5	mae-P3.23 257	. 2 ⊢ (𝛾 → (𝑡 = 𝑢 → 𝑢 = 𝑡))
7	1, 6	ndime-P3.6 171	1 ⊢ (𝛾 → 𝑢 = 𝑡)

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:  eqsym-P5.CL  628  subeql-P5  632  spliteq-P6-L1  775  psubsuccv-P6-L1  805  psubaddv-P6-L1  807  psubmultv-P6-L1  809