Theorem nfre1 2407

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 2354	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1425	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1403	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 102  Ⅎwnf 1389  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-rex 2354

This theorem is referenced by:  nfiu1  3708  fun11iun  5167  eusvobj2  5518  prarloclem3step  6686  prmuloc2  6757  ltexprlemm  6790  caucvgprprlemaddq  6898  caucvgsrlemgt1  6971  supinfneg  8683  infsupneg  8684  lbzbi  8701  divalglemeunn  10321  divalglemeuneg  10323  bezoutlemmain  10387  bezout  10400