Theorem eqsym-P7 936

Description: Equivalence Property: '=' Symmetry. †

Hypothesis

Ref	Expression
eqsym-P7.1	⊢ (𝛾 → 𝑡 = 𝑢)

Assertion

Ref	Expression
eqsym-P7	⊢ (𝛾 → 𝑢 = 𝑡)

Proof of Theorem eqsym-P7

Step	Hyp	Ref	Expression
1		ndeqi-P7.21 846	. . 3 ⊢ 𝑡 = 𝑡
2	1	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → 𝑡 = 𝑡)
3		eqsym-P7.1	. . 3 ⊢ (𝛾 → 𝑡 = 𝑢)
4	3	ndsubeql-P7.22a 847	. 2 ⊢ (𝛾 → (𝑡 = 𝑡 ↔ 𝑢 = 𝑡))
5	2, 4	bimpf-P4 531	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-P7.CL  937  eqsym-P7r  983  eqsym-P7r.RC  984