Theorem 3eltr3d 2279

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

Hypotheses

Ref	Expression
3eltr3d.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
3eltr3d.2	⊢ (𝜑 → 𝐴 = 𝐶)
3eltr3d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3eltr3d	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Proof of Theorem 3eltr3d

Step	Hyp	Ref	Expression
1		3eltr3d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
2		3eltr3d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		3eltr3d.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 3	eleqtrd 2275	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
5	1, 4	eqeltrrd 2274	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  reg3exmidlemwe  4616  nnaordi  6575  icoshftf1o  10083  lincmb01cmp  10095  fzosubel  10287  cnmpt2res  14617  dvcnp2cntop  15019