Theorem 3eltr3g 2122

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 2105	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐶 ∈ 𝐷)
5	1, 4	sylib 127	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by: (None)