Theorem subelofl-P5 638

Description: Left Substitution for '∈'.

Hypothesis

Ref	Expression
subelofl-P5.1	⊢ (𝛾 → 𝑡 = 𝑢)

Assertion

Ref	Expression
subelofl-P5	⊢ (𝛾 → (𝑡 ∈ 𝑤 ↔ 𝑢 ∈ 𝑤))

Proof of Theorem subelofl-P5

Step	Hyp	Ref	Expression
1		subelofl-P5.1	. 2 ⊢ (𝛾 → 𝑡 = 𝑢)
2		ax-L8-inl 20	. . 3 ⊢ (𝑡 = 𝑢 → (𝑡 ∈ 𝑤 → 𝑢 ∈ 𝑤))
3		ax-L8-inl 20	. . . 4 ⊢ (𝑢 = 𝑡 → (𝑢 ∈ 𝑤 → 𝑡 ∈ 𝑤))
4		eqsym-P5.CL.SYM 629	. . . 4 ⊢ (𝑢 = 𝑡 ↔ 𝑡 = 𝑢)
5	3, 4	subiml2-P4.RC 541	. . 3 ⊢ (𝑡 = 𝑢 → (𝑢 ∈ 𝑤 → 𝑡 ∈ 𝑤))
6	2, 5	ndbii-P3.13 178	. 2 ⊢ (𝑡 = 𝑢 → (𝑡 ∈ 𝑤 ↔ 𝑢 ∈ 𝑤))
7	1, 6	syl-P3.24.RC 260	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:  subelofl-P5.CL  639  subelofd-P5  642  ndsubelofl-P7.23a  849