Theorem ndsubelofd-P7.CL 916

Description: Closed Form of ndsubelofd-P7 857. †

Assertion

Ref	Expression
ndsubelofd-P7.CL	⊢ ((𝑠 = 𝑡 ∧ 𝑢 = 𝑤) → (𝑠 ∈ 𝑢 ↔ 𝑡 ∈ 𝑤))

Proof of Theorem ndsubelofd-P7.CL

Step	Hyp	Ref	Expression
1		rcp-NDASM1of2 193	. 2 ⊢ ((𝑠 = 𝑡 ∧ 𝑢 = 𝑤) → 𝑠 = 𝑡)
2		rcp-NDASM2of2 194	. 2 ⊢ ((𝑠 = 𝑡 ∧ 𝑢 = 𝑤) → 𝑢 = 𝑤)
3	1, 2	ndsubelofd-P7 857	1 ⊢ ((𝑠 = 𝑡 ∧ 𝑢 = 𝑤) → (𝑠 ∈ 𝑢 ↔ 𝑡 ∈ 𝑤))

Syntax hints:   = wff-equals 6   ∈ wff-elemof 7   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132

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  ax-L8-inl 20  ax-L8-inr 21

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: (None)