Theorem nfre1 2587

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 2528	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1545	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1523	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1509  ∃wex 1541   ∈ wcel 2205  ∃wrex 2523

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 1496  ax-gen 1498  ax-ie1 1542

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-rex 2528

This theorem is referenced by:  r19.29an  2687  nfiu1  4026  fun11iun  5640  eusvobj2  6044  fodjuomnilemdc  7448  ismkvnex  7459  prarloclem3step  7827  prmuloc2  7898  ltexprlemm  7931  caucvgprprlemaddq  8039  caucvgsrlemgt1  8126  axpre-suploclemres  8232  supinfneg  9945  infsupneg  9946  lbzbi  9966  divalglemeunn  12632  divalglemeuneg  12634  bezoutlemmain  12719  bezout  12732  lss1d  14657  pw1nct  16903  isomninnlem  16940  trirec0  16954  ismkvnnlem  16963