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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 1811 | . . 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 7137 | . 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)) |
7 | 6 | mptru 1549 | 1 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1543 Ⅎwnf 1790 Ⅎwnfc 2879 ℩crio 7128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3400 df-in 3850 df-ss 3860 df-sn 4517 df-uni 4797 df-iota 6297 df-riota 7129 |
This theorem is referenced by: csbriota 7145 nfoi 9053 lble 11672 nosupbnd1 33562 noinfbnd1 33577 riotasvd 36615 riotasv2d 36616 riotasv2s 36617 cdleme26ee 38019 cdleme31sn1 38040 cdlemefs32sn1aw 38073 cdleme43fsv1snlem 38079 cdleme41sn3a 38092 cdleme32d 38103 cdleme32f 38105 cdleme40m 38126 cdleme40n 38127 cdlemk36 38572 cdlemk38 38574 cdlemkid 38595 cdlemk19x 38602 cdlemk11t 38605 |
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