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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 1804 | . . 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 1547 | 1 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊤wtru 1541 Ⅎwnf 1783 Ⅎwnfc 2890 ℩crio 7387 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-sn 4627 df-uni 4908 df-iota 6514 df-riota 7388 | 
| This theorem is referenced by: csbriota 7403 nfoi 9554 lble 12220 nosupbnd1 27759 noinfbnd1 27774 riotasvd 38957 riotasv2d 38958 riotasv2s 38959 cdleme26ee 40362 cdleme31sn1 40383 cdlemefs32sn1aw 40416 cdleme43fsv1snlem 40422 cdleme41sn3a 40435 cdleme32d 40446 cdleme32f 40448 cdleme40m 40469 cdleme40n 40470 cdlemk36 40915 cdlemk38 40917 cdlemkid 40938 cdlemk19x 40945 cdlemk11t 40948 | 
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