Theorem nfbi 1642

Description: If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ↔ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

Ref	Expression
nfbi.1	⊢ Ⅎ𝑥𝜑
nfbi.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
nfbi	⊢ Ⅎ𝑥(𝜑 ↔ 𝜓)

Proof of Theorem nfbi

Step	Hyp	Ref	Expression
1		nfbi.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
3		nfbi.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜓)
5	2, 4	nfbid 1641	. 2 ⊢ (⊤ → Ⅎ𝑥(𝜑 ↔ 𝜓))
6	5	mptru 1411	1 ⊢ Ⅎ𝑥(𝜑 ↔ 𝜓)

Syntax hints:   ↔ wb 105  ⊤wtru 1403  Ⅎwnf 1513

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 1500  ax-gen 1502  ax-4 1563  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514

This theorem is referenced by:  sb8eu  2099  nfeuv  2104  bm1.1  2223  abbibcom  2352  abbib  2356  nfeq  2400  cleqf  2417  sbhypf  2872  ceqsexg  2954  elabgt  2967  elabgf  2968  copsex2t  4380  copsex2g  4381  opelopabsb  4397  opeliunxp2  4915  ralxpf  4921  rexxpf  4922  cbviota  5337  sb8iota  5340  fmptco  5865  nfiso  6002  uchoice  6361  dfoprab4f  6417  opeliunxp2f  6499  xpf1o  7134  bdsepnfALT  16829