Theorem nfre1 2575

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 2516	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1544	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1522	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1508  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-rex 2516

This theorem is referenced by:  r19.29an  2675  nfiu1  4000  fun11iun  5604  eusvobj2  6003  fodjuomnilemdc  7342  ismkvnex  7353  prarloclem3step  7715  prmuloc2  7786  ltexprlemm  7819  caucvgprprlemaddq  7927  caucvgsrlemgt1  8014  axpre-suploclemres  8120  supinfneg  9828  infsupneg  9829  lbzbi  9849  divalglemeunn  12481  divalglemeuneg  12483  bezoutlemmain  12568  bezout  12581  lss1d  14396  pw1nct  16604  isomninnlem  16634  trirec0  16648  ismkvnnlem  16656