Theorem nfre1 2573

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 2514	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1542	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1520	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1506  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-rex 2514

This theorem is referenced by:  r19.29an  2673  nfiu1  3994  fun11iun  5592  eusvobj2  5986  fodjuomnilemdc  7307  ismkvnex  7318  prarloclem3step  7679  prmuloc2  7750  ltexprlemm  7783  caucvgprprlemaddq  7891  caucvgsrlemgt1  7978  axpre-suploclemres  8084  supinfneg  9786  infsupneg  9787  lbzbi  9807  divalglemeunn  12427  divalglemeuneg  12429  bezoutlemmain  12514  bezout  12527  lss1d  14341  pw1nct  16328  isomninnlem  16357  trirec0  16371  ismkvnnlem  16379