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  6004  fodjuomnilemdc  7343  ismkvnex  7354  prarloclem3step  7716  prmuloc2  7787  ltexprlemm  7820  caucvgprprlemaddq  7928  caucvgsrlemgt1  8015  axpre-suploclemres  8121  supinfneg  9829  infsupneg  9830  lbzbi  9850  divalglemeunn  12487  divalglemeuneg  12489  bezoutlemmain  12574  bezout  12587  lss1d  14403  pw1nct  16630  isomninnlem  16660  trirec0  16674  ismkvnnlem  16683