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Mirrors > Home > ILE Home > Th. List > nfbi | GIF version |
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.) |
Ref | Expression |
---|---|
nfbi.1 | ⊢ Ⅎ𝑥𝜑 |
nfbi.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
nfbi | ⊢ Ⅎ𝑥(𝜑 ↔ 𝜓) |
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 1588 | . 2 ⊢ (⊤ → Ⅎ𝑥(𝜑 ↔ 𝜓)) |
6 | 5 | mptru 1362 | 1 ⊢ Ⅎ𝑥(𝜑 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ⊤wtru 1354 Ⅎwnf 1460 |
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 1447 ax-gen 1449 ax-4 1510 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 |
This theorem is referenced by: sb8eu 2039 nfeuv 2044 bm1.1 2162 abbi 2291 nfeq 2327 cleqf 2344 sbhypf 2787 ceqsexg 2866 elabgt 2879 elabgf 2880 copsex2t 4246 copsex2g 4247 opelopabsb 4261 opeliunxp2 4768 ralxpf 4774 rexxpf 4775 cbviota 5184 sb8iota 5186 fmptco 5683 nfiso 5807 dfoprab4f 6194 opeliunxp2f 6239 xpf1o 6844 bdsepnfALT 14644 |
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