Theorem nfre1 2548

Description: 𝑥 is not free in ∃𝑥 ∈ 𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)

Assertion

Ref	Expression
nfre1	⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Proof of Theorem nfre1

Step	Hyp	Ref	Expression
1		df-rex 2489	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1518	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1496	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1482  ∃wex 1514   ∈ wcel 2175  ∃wrex 2484

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-5 1469  ax-gen 1471  ax-ie1 1515

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-rex 2489

This theorem is referenced by:  r19.29an  2647  nfiu1  3956  fun11iun  5542  eusvobj2  5929  fodjuomnilemdc  7245  ismkvnex  7256  prarloclem3step  7608  prmuloc2  7679  ltexprlemm  7712  caucvgprprlemaddq  7820  caucvgsrlemgt1  7907  axpre-suploclemres  8013  supinfneg  9715  infsupneg  9716  lbzbi  9736  divalglemeunn  12174  divalglemeuneg  12176  bezoutlemmain  12261  bezout  12274  lss1d  14087  pw1nct  15873  isomninnlem  15902  trirec0  15916  ismkvnnlem  15924