Theorem nfre1 2474

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 2420	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1472	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1450	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 103  Ⅎwnf 1436  ∃wex 1468   ∈ wcel 1480  ∃wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-rex 2420

This theorem is referenced by:  r19.29an  2572  nfiu1  3838  fun11iun  5381  eusvobj2  5753  fodjuomnilemdc  7009  ismkvnex  7022  prarloclem3step  7297  prmuloc2  7368  ltexprlemm  7401  caucvgprprlemaddq  7509  caucvgsrlemgt1  7596  axpre-suploclemres  7702  supinfneg  9383  infsupneg  9384  lbzbi  9401  divalglemeunn  11607  divalglemeuneg  11609  bezoutlemmain  11675  bezout  11688  isomninnlem  13214