Theorem 3eltr3g 2222

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

Hypotheses

Ref	Expression
3eltr3g.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
3eltr3g.2	⊢ 𝐴 = 𝐶
3eltr3g.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3eltr3g	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Proof of Theorem 3eltr3g

Step	Hyp	Ref	Expression
1		3eltr3g.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		3eltr3g.2	. . 3 ⊢ 𝐴 = 𝐶
3		3eltr3g.3	. . 3 ⊢ 𝐵 = 𝐷
4	2, 3	eleq12i 2205	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐶 ∈ 𝐷)
5	1, 4	sylib 121	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133

This theorem is referenced by: (None)