Theorem iineq1d 39267

Description: Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypothesis

Ref	Expression
iineq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
iineq1d	|- ( ph -> |^|_ x e. A C = |^|_ x e. B C )

Distinct variable groups:   x, A   x, B

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

Proof of Theorem iineq1d

Step	Hyp	Ref	Expression
1		iineq1d.1	. 2 |- ( ph -> A = B )
2		iineq1 4535	. 2 |- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )
3	1, 2	syl 17	1 |- ( ph -> |^|_ x e. A C = |^|_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1483   |^|_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:  iineq12dv  39289  smflimlem2  40980  smflimlem3  40981  smflimlem4  40982  smflim2  41012  smflimsuplem1  41026  smflimsuplem7  41032  smflimsup  41034  smfliminf  41037