Theorem raleqtrrdv 2713

Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)

Hypotheses

Ref	Expression
raleqtrrdv.1	|- ( ph -> A. x e. A ps )
raleqtrrdv.2	|- ( ph -> B = A )

Assertion

Ref	Expression
raleqtrrdv	|- ( ph -> A. x e. B ps )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem raleqtrrdv

Step	Hyp	Ref	Expression
1		raleqtrrdv.1	. 2 |- ( ph -> A. x e. A ps )
2		raleqtrrdv.2	. . 3 |- ( ph -> B = A )
3	2	raleqdv 2709	. 2 |- ( ph -> ( A. x e. B ps <-> A. x e. A ps ) )
4	1, 3	mpbird 167	1 |- ( ph -> A. x e. B ps )

Syntax hints:   -> wi 4   = wceq 1373   A.wral 2485

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490

This theorem is referenced by: (None)