Theorem raleleqALT 3157

Description: Alternate proof of raleleq 3156 using ralel 2923, being longer and using more axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
raleleqALT	|- ( A = B -> A. x e. A x e. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem raleleqALT

Step	Hyp	Ref	Expression
1		ralel 2923	. 2 |- A. x e. B x e. B
2		id 22	. . 3 |- ( A = B -> A = B )
3	2	raleqdv 3144	. 2 |- ( A = B -> ( A. x e. A x e. B <-> A. x e. B x e. B ) )
4	1, 3	mpbiri 248	1 |- ( A = B -> A. x e. A x e. B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912

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-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by: (None)