Theorem ndsubelofl-P7.23a 849

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

Hypothesis

Ref	Expression
ndsubelofl-P7.23a.1	⊢ (𝛾 → 𝑡 = 𝑢)

Assertion

Ref	Expression
ndsubelofl-P7.23a	⊢ (𝛾 → (𝑡 ∈ 𝑤 ↔ 𝑢 ∈ 𝑤))

Proof of Theorem ndsubelofl-P7.23a

Step	Hyp	Ref	Expression
1		ndsubelofl-P7.23a.1	. 2 ⊢ (𝛾 → 𝑡 = 𝑢)
2	1	subelofl-P5 638	1 ⊢ (𝛾 → (𝑡 ∈ 𝑤 ↔ 𝑢 ∈ 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

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  857  ndsubelofl-P7.23a.RC  894  ndsubelofl-P7.23a.CL  914