Theorem 3eltr4g 2226

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
3eltr4g.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
3eltr4g.2	⊢ 𝐶 = 𝐴
3eltr4g.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3eltr4g	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Proof of Theorem 3eltr4g

Step	Hyp	Ref	Expression
1		3eltr4g.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		3eltr4g.2	. . 3 ⊢ 𝐶 = 𝐴
3		3eltr4g.3	. . 3 ⊢ 𝐷 = 𝐵
4	2, 3	eleq12i 2208	. 2 ⊢ (𝐶 ∈ 𝐷 ↔ 𝐴 ∈ 𝐵)
5	1, 4	sylibr 133	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136

This theorem is referenced by:  riotacl2  5751  2strop1g  12103