Theorem 3eltr4g 2856

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)

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.2	. . 3 ⊢ 𝐶 = 𝐴
2		3eltr4g.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3	1, 2	syl5eqel 2843	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
4		3eltr4g.3	. 2 ⊢ 𝐷 = 𝐵
5	3, 4	syl6eleqr 2850	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2753  df-clel 2756

This theorem is referenced by:  riotacl2  6788  rankelun  8910  rankelpr  8911  rankelop  8912  cdivcncf  22941  itg1addlem4  23685  cxpcn3  24709  bposlem4  25232  mirauto  25799  ldgenpisyslem1  30556  nosepdm  32161  relowlpssretop  33541  mapfzcons  37799  fourierdlem62  40906  fourierdlem63  40907