Theorem ndsubelofd-P7 857

Description: Natural Deduction: Predicate Substitution Rule ('∈' dual). †

Hypotheses

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

Assertion

Ref	Expression
ndsubelofd-P7	⊢ (𝛾 → (𝑠 ∈ 𝑢 ↔ 𝑡 ∈ 𝑤))

Proof of Theorem ndsubelofd-P7

Step	Hyp	Ref	Expression
1		ndsubelofd-P7.1	. . 3 ⊢ (𝛾 → 𝑠 = 𝑡)
2	1	ndsubelofl-P7.23a 849	. 2 ⊢ (𝛾 → (𝑠 ∈ 𝑢 ↔ 𝑡 ∈ 𝑢))
3		ndsubelofd-P7.2	. . 3 ⊢ (𝛾 → 𝑢 = 𝑤)
4	3	ndsubelofr-P7.23b 850	. 2 ⊢ (𝛾 → (𝑡 ∈ 𝑢 ↔ 𝑡 ∈ 𝑤))
5	2, 4	bitrns-P3.33c 302	1 ⊢ (𝛾 → (𝑠 ∈ 𝑢 ↔ 𝑡 ∈ 𝑤))

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

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:  ndsubelofd-P7.RC  896  ndsubelofd-P7.CL  916