Theorem nfre1 2518

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 2459	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfe1 1494	. 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)
3	1, 2	nfxfr 1472	1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑

Syntax hints:   ∧ wa 104  Ⅎwnf 1458  ∃wex 1490   ∈ wcel 2146  ∃wrex 2454

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 1445  ax-gen 1447  ax-ie1 1491

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-rex 2459

This theorem is referenced by:  r19.29an  2617  nfiu1  3912  fun11iun  5474  eusvobj2  5851  fodjuomnilemdc  7132  ismkvnex  7143  prarloclem3step  7470  prmuloc2  7541  ltexprlemm  7574  caucvgprprlemaddq  7682  caucvgsrlemgt1  7769  axpre-suploclemres  7875  supinfneg  9566  infsupneg  9567  lbzbi  9587  divalglemeunn  11891  divalglemeuneg  11893  bezoutlemmain  11964  bezout  11977  pw1nct  14293  isomninnlem  14319  trirec0  14333  ismkvnnlem  14341