Theorem nfre1 2533

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 2474	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1507	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1485	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1471  ∃wex 1503   ∈ wcel 2160  ∃wrex 2469

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 1458  ax-gen 1460  ax-ie1 1504

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-rex 2474

This theorem is referenced by:  r19.29an  2632  nfiu1  3931  fun11iun  5501  eusvobj2  5881  fodjuomnilemdc  7171  ismkvnex  7182  prarloclem3step  7524  prmuloc2  7595  ltexprlemm  7628  caucvgprprlemaddq  7736  caucvgsrlemgt1  7823  axpre-suploclemres  7929  supinfneg  9624  infsupneg  9625  lbzbi  9645  divalglemeunn  11957  divalglemeuneg  11959  bezoutlemmain  12030  bezout  12043  lss1d  13696  pw1nct  15206  isomninnlem  15232  trirec0  15246  ismkvnnlem  15254