Theorem nfre1 2550

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 2491	. 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 2177  ∃wrex 2486

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 2491

This theorem is referenced by:  r19.29an  2649  nfiu1  3963  fun11iun  5555  eusvobj2  5943  fodjuomnilemdc  7261  ismkvnex  7272  prarloclem3step  7629  prmuloc2  7700  ltexprlemm  7733  caucvgprprlemaddq  7841  caucvgsrlemgt1  7928  axpre-suploclemres  8034  supinfneg  9736  infsupneg  9737  lbzbi  9757  divalglemeunn  12307  divalglemeuneg  12309  bezoutlemmain  12394  bezout  12407  lss1d  14220  pw1nct  16081  isomninnlem  16110  trirec0  16124  ismkvnnlem  16132