Theorem nfre1 2585

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 2526	. 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 2203  ∃wrex 2521

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 2526

This theorem is referenced by:  r19.29an  2685  nfiu1  4021  fun11iun  5635  eusvobj2  6036  fodjuomnilemdc  7435  ismkvnex  7446  prarloclem3step  7811  prmuloc2  7882  ltexprlemm  7915  caucvgprprlemaddq  8023  caucvgsrlemgt1  8110  axpre-suploclemres  8216  supinfneg  9927  infsupneg  9928  lbzbi  9948  divalglemeunn  12607  divalglemeuneg  12609  bezoutlemmain  12694  bezout  12707  lss1d  14531  pw1nct  16777  isomninnlem  16814  trirec0  16828  ismkvnnlem  16837