Theorem inteqd 1970

Description: Equality deduction for class intersection.

Hypothesis

Ref	Expression
inteqd.1	⊢ (φ → A = B)

Assertion

Ref	Expression
inteqd	⊢ (φ → ∩A = ∩B)

Proof of Theorem inteqd

Step	Hyp	Ref	Expression
1		inteqd.1	. 2 ⊢ (φ → A = B)
2		inteq 1968	. 2 ⊢ (A = B → ∩A = ∩B)
3	1, 2	syl 12	1 ⊢ (φ → ∩A = ∩B)

Syntax hints:   → wi 2   = wceq 1091  ∩cint 1965

This theorem is referenced by:  onsucmin 2323  elreldm 2554  elxp5 2641  fundmen 3333  xpsnen 3339  mapunen 3397  unblem2 3432  unblem3 3433  fiint 3445  tz9.12lem1 3503  tz9.12lem3 3505  rankval 3512  rankvalg 3513  rankonid 3538  oncardval 3626  cardval 3633  alephon 3671  alephsuc 3672  cfval 3701  xpnnen 4927  spanvalt 5300  hsupval2t 5301  chsupid 5312

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-int 1966

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