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Mirrors > Home > ILE Home > Th. List > nfra1 | GIF version |
Description: 𝑥 is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfra1 | ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2460 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | nfa1 1541 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐴 → 𝜑) | |
3 | 1, 2 | nfxfr 1474 | 1 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 Ⅎwnf 1460 ∈ wcel 2148 ∀wral 2455 |
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-ial 1534 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-ral 2460 |
This theorem is referenced by: nfra2xy 2519 r19.12 2583 ralbi 2609 rexbi 2610 nfss 3149 ralidm 3524 nfii1 3918 dfiun2g 3919 mpteq12f 4084 reusv1 4459 ralxfrALT 4468 peano2 4595 fun11iun 5483 fvmptssdm 5601 ffnfv 5675 riota5f 5855 mpoeq123 5934 tfri3 6368 nfixp1 6718 nneneq 6857 exmidomni 7140 mkvprop 7156 caucvgsrlemgt1 7794 suplocsrlem 7807 lble 8904 indstr 9593 fimaxre2 11235 prodeq2 11565 zsupcllemstep 11946 bezoutlemmain 11999 bezoutlemzz 12003 exmidunben 12427 mulcncf 14094 limccnp2cntop 14149 bj-rspgt 14541 isomninnlem 14781 iswomninnlem 14800 ismkvnnlem 14803 |
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