Theorem iineq12dv 39289

Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
iineq12dv.1	|- ( ph -> A = B )
iineq12dv.2	|- ( ( ph /\ x e. B ) -> C = D )

Assertion

Ref	Expression
iineq12dv	|- ( ph -> |^|_ x e. A C = |^|_ x e. B D )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   C( x)   D( x)

Proof of Theorem iineq12dv

Step	Hyp	Ref	Expression
1		iineq12dv.1	. . 3 |- ( ph -> A = B )
2	1	iineq1d 39267	. 2 |- ( ph -> |^|_ x e. A C = |^|_ x e. B C )
3		iineq12dv.2	. . 3 |- ( ( ph /\ x e. B ) -> C = D )
4	3	iineq2dv 4543	. 2 |- ( ph -> |^|_ x e. B C = |^|_ x e. B D )
5	2, 4	eqtrd 2656	1 |- ( ph -> |^|_ x e. A C = |^|_ x e. B D )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |^|_ciin 4521

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-13 2246  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-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-iin 4523

This theorem is referenced by:  smflim  40985