Theorem nfre1 2551

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 2492	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1520	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1498	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1484  ∃wex 1516   ∈ wcel 2178  ∃wrex 2487

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 1471  ax-gen 1473  ax-ie1 1517

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-rex 2492

This theorem is referenced by:  r19.29an  2650  nfiu1  3971  fun11iun  5565  eusvobj2  5953  fodjuomnilemdc  7272  ismkvnex  7283  prarloclem3step  7644  prmuloc2  7715  ltexprlemm  7748  caucvgprprlemaddq  7856  caucvgsrlemgt1  7943  axpre-suploclemres  8049  supinfneg  9751  infsupneg  9752  lbzbi  9772  divalglemeunn  12347  divalglemeuneg  12349  bezoutlemmain  12434  bezout  12447  lss1d  14260  pw1nct  16142  isomninnlem  16171  trirec0  16185  ismkvnnlem  16193