Theorem nfre1 2593

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 2534	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1549	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1527	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1513  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529

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 1500  ax-gen 1502  ax-ie1 1546

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-rex 2534

This theorem is referenced by:  r19.29an  2693  nfiu1  4037  fun11iun  5655  eusvobj2  6061  fodjuomnilemdc  7474  ismkvnex  7485  prarloclem3step  7853  prmuloc2  7924  ltexprlemm  7957  caucvgprprlemaddq  8065  caucvgsrlemgt1  8152  axpre-suploclemres  8258  supinfneg  9974  infsupneg  9975  lbzbi  9995  divalglemeunn  12666  divalglemeuneg  12668  bezoutlemmain  12753  bezout  12766  lss1d  14692  pw1nct  16947  isomninnlem  16984  trirec0  16998  ismkvnnlem  17007