Theorem eqtrns-P7 987

Description: Equivalence Property: '=' Transitivity. †

Hypotheses

Ref	Expression
eqtrns-P7.1	⊢ (𝛾 → 𝑡 = 𝑢)
eqtrns-P7.2	⊢ (𝛾 → 𝑢 = 𝑤)

Assertion

Ref	Expression
eqtrns-P7	⊢ (𝛾 → 𝑡 = 𝑤)

Proof of Theorem eqtrns-P7

Step	Hyp	Ref	Expression
1		eqtrns-P7.1	. 2 ⊢ (𝛾 → 𝑡 = 𝑢)
2		eqtrns-P7.2	. . 3 ⊢ (𝛾 → 𝑢 = 𝑤)
3	2	ndsubeqr-P7.22b 848	. 2 ⊢ (𝛾 → (𝑡 = 𝑢 ↔ 𝑡 = 𝑤))
4	1, 3	bimpf-P4 531	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-P7.RC  988  eqtrns-P7.CL  989  example-E7.1a  1074  example-E7.1b  1075