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Mirrors > Home > MPE Home > Th. List > nfriota | Structured version Visualization version GIF version |
Description: A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
nfriota.1 | ⊢ Ⅎ𝑥𝜑 |
nfriota.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfriota | ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1801 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfriota.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
4 | nfriota.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
6 | 1, 3, 5 | nfriotadw 7396 | . 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)) |
7 | 6 | mptru 1544 | 1 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1538 Ⅎwnf 1780 Ⅎwnfc 2888 ℩crio 7387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-v 3480 df-ss 3980 df-sn 4632 df-uni 4913 df-iota 6516 df-riota 7388 |
This theorem is referenced by: csbriota 7403 nfoi 9552 lble 12218 nosupbnd1 27774 noinfbnd1 27789 riotasvd 38938 riotasv2d 38939 riotasv2s 38940 cdleme26ee 40343 cdleme31sn1 40364 cdlemefs32sn1aw 40397 cdleme43fsv1snlem 40403 cdleme41sn3a 40416 cdleme32d 40427 cdleme32f 40429 cdleme40m 40450 cdleme40n 40451 cdlemk36 40896 cdlemk38 40898 cdlemkid 40919 cdlemk19x 40926 cdlemk11t 40929 |
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