Theorem nfre1 2537

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 2478	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1507	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1485	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1471  ∃wex 1503   ∈ wcel 2164  ∃wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-rex 2478

This theorem is referenced by:  r19.29an  2636  nfiu1  3942  fun11iun  5521  eusvobj2  5904  fodjuomnilemdc  7203  ismkvnex  7214  prarloclem3step  7556  prmuloc2  7627  ltexprlemm  7660  caucvgprprlemaddq  7768  caucvgsrlemgt1  7855  axpre-suploclemres  7961  supinfneg  9660  infsupneg  9661  lbzbi  9681  divalglemeunn  12062  divalglemeuneg  12064  bezoutlemmain  12135  bezout  12148  lss1d  13879  pw1nct  15493  isomninnlem  15520  trirec0  15534  ismkvnnlem  15542