Theorem 3eltr3g 2717

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
3eltr3g.1	|- ( ph -> A e. B )
3eltr3g.2	|- A = C
3eltr3g.3	|- B = D

Assertion

Ref	Expression
3eltr3g	|- ( ph -> C e. D )

Proof of Theorem 3eltr3g

Step	Hyp	Ref	Expression
1		3eltr3g.2	. . 3 |- A = C
2		3eltr3g.1	. . 3 |- ( ph -> A e. B )
3	1, 2	syl5eqelr 2706	. 2 |- ( ph -> C e. B )
4		3eltr3g.3	. 2 |- B = D
5	3, 4	syl6eleq 2711	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by:  rankelpr  8736  isf34lem7  9201  rmulccn  29974  xrge0mulc1cn  29987  esumpfinvallem  30136  fourierdlem62  40385