Theorem nfre1 2513

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 2454	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1489	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1467	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 103  Ⅎwnf 1453  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449

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 1440  ax-gen 1442  ax-ie1 1486

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-rex 2454

This theorem is referenced by:  r19.29an  2612  nfiu1  3903  fun11iun  5463  eusvobj2  5839  fodjuomnilemdc  7120  ismkvnex  7131  prarloclem3step  7458  prmuloc2  7529  ltexprlemm  7562  caucvgprprlemaddq  7670  caucvgsrlemgt1  7757  axpre-suploclemres  7863  supinfneg  9554  infsupneg  9555  lbzbi  9575  divalglemeunn  11880  divalglemeuneg  11882  bezoutlemmain  11953  bezout  11966  pw1nct  14036  isomninnlem  14062  trirec0  14076  ismkvnnlem  14084