Theorem inteqd 3975

Description: Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)

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 3973	. 2 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1402  ∩ cint 3970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-int 3971

This theorem is used by:  intprg  4003  op1stbg  4627  onsucmin  4656  elreldm  5010  elxp5  5279  fniinfv  5764  1stval2  6393  2ndval2  6394  fundmen  7099  xpsnen  7124  fiintim  7243  elfi2  7312  fi0  7315  cardcl  7533  isnumi  7534  cardval3ex  7537  carden2bex  7542  lspfval  14835  lspval  14837  lsppropd  14879  aspval  15125  clsfval  15339  clsval  15349