Theorem nfbid 1905

Description: If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ↔ 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)

Hypotheses

Ref	Expression
nfbid.1	⊢ (𝜑 → Ⅎ𝑥𝜓)
nfbid.2	⊢ (𝜑 → Ⅎ𝑥𝜒)

Assertion

Ref	Expression
nfbid	⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒))

Proof of Theorem nfbid

Step	Hyp	Ref	Expression
1		dfbi2 475	. 2 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
2		nfbid.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		nfbid.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	2, 3	nfimd 1897	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))
5	3, 2	nfimd 1897	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜒 → 𝜓))
6	4, 5	nfand 1900	. 2 ⊢ (𝜑 → Ⅎ𝑥((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
7	1, 6	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  nfbi  1906  nfeqd  2917  nfiotadw  6394  nfiotad  6396  iota2df  6420  axextnd  10347  axrepndlem1  10348  axrepndlem2  10349  axacndlem4  10366  axacndlem5  10367  axacnd  10368  axextdist  33775  copsex2d  35310  cbveud  35543  wl-eudf  35727  wl-sb8eut  35732