Theorem subelofr-P5 640

Description: Right Substitution for '∈'.

Hypothesis

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

Assertion

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

Proof of Theorem subelofr-P5

Step	Hyp	Ref	Expression
1		subelofr-P5.1	. 2 ⊢ (𝛾 → 𝑡 = 𝑢)
2		ax-L8-inr 21	. . 3 ⊢ (𝑡 = 𝑢 → (𝑤 ∈ 𝑡 → 𝑤 ∈ 𝑢))
3		ax-L8-inr 21	. . . 4 ⊢ (𝑢 = 𝑡 → (𝑤 ∈ 𝑢 → 𝑤 ∈ 𝑡))
4		eqsym-P5.CL.SYM 629	. . . 4 ⊢ (𝑢 = 𝑡 ↔ 𝑡 = 𝑢)
5	3, 4	subiml2-P4.RC 541	. . 3 ⊢ (𝑡 = 𝑢 → (𝑤 ∈ 𝑢 → 𝑤 ∈ 𝑡))
6	2, 5	ndbii-P3.13 178	. 2 ⊢ (𝑡 = 𝑢 → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ 𝑢))
7	1, 6	syl-P3.24.RC 260	1 ⊢ (𝛾 → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ 𝑢))

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-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:  subelofr-P5.CL  641  subelofd-P5  642  example-E5.03a  665  ndsubelofr-P7.23b  850