Theorem ndsubelofd-P7.RC 896

Description: Inference Form of ndsubelofd-P7 857. †

Hypotheses

Ref	Expression
ndsubelofd-P7.RC.1	⊢ 𝑠 = 𝑡
ndsubelofd-P7.RC.2	⊢ 𝑢 = 𝑤

Assertion

Ref	Expression
ndsubelofd-P7.RC	⊢ (𝑠 ∈ 𝑢 ↔ 𝑡 ∈ 𝑤)

Proof of Theorem ndsubelofd-P7.RC

Step	Hyp	Ref	Expression
1		ndsubelofd-P7.RC.1	. . . 4 ⊢ 𝑠 = 𝑡
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → 𝑠 = 𝑡)
3		ndsubelofd-P7.RC.2	. . . 4 ⊢ 𝑢 = 𝑤
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → 𝑢 = 𝑤)
5	2, 4	ndsubelofd-P7 857	. 2 ⊢ (⊤ → (𝑠 ∈ 𝑢 ↔ 𝑡 ∈ 𝑤))
6	5	ndtruee-P3.18 183	1 ⊢ (𝑠 ∈ 𝑢 ↔ 𝑡 ∈ 𝑤)

Syntax hints:   = wff-equals 6   ∈ wff-elemof 7   ↔ wff-bi 104  ⊤wff-true 153

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)