Theorem nfre1 2520

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 2461	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1496	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1474	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1460  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-rex 2461

This theorem is referenced by:  r19.29an  2619  nfiu1  3918  fun11iun  5484  eusvobj2  5863  fodjuomnilemdc  7144  ismkvnex  7155  prarloclem3step  7497  prmuloc2  7568  ltexprlemm  7601  caucvgprprlemaddq  7709  caucvgsrlemgt1  7796  axpre-suploclemres  7902  supinfneg  9597  infsupneg  9598  lbzbi  9618  divalglemeunn  11928  divalglemeuneg  11930  bezoutlemmain  12001  bezout  12014  lss1d  13475  pw1nct  14837  isomninnlem  14863  trirec0  14877  ismkvnnlem  14885