Theorem nfre1 2540

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 2481	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1510	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1488	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1474  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-rex 2481

This theorem is referenced by:  r19.29an  2639  nfiu1  3946  fun11iun  5525  eusvobj2  5908  fodjuomnilemdc  7210  ismkvnex  7221  prarloclem3step  7563  prmuloc2  7634  ltexprlemm  7667  caucvgprprlemaddq  7775  caucvgsrlemgt1  7862  axpre-suploclemres  7968  supinfneg  9669  infsupneg  9670  lbzbi  9690  divalglemeunn  12086  divalglemeuneg  12088  bezoutlemmain  12165  bezout  12178  lss1d  13939  pw1nct  15647  isomninnlem  15674  trirec0  15688  ismkvnnlem  15696