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Mirrors > Home > ILE Home > Th. List > spcgv | GIF version |
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) |
Ref | Expression |
---|---|
spcgv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcgv | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2319 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1528 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | spcgv.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | spcgf 2819 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 = wceq 1353 ∈ wcel 2148 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 |
This theorem is referenced by: spcv 2831 mob2 2917 intss1 3859 dfiin2g 3919 exmidsssnc 4203 exmid1stab 4208 frirrg 4350 frind 4352 alxfr 4461 elirr 4540 en2lp 4553 tfisi 4586 mptfvex 5601 tfrcl 6364 rdgisucinc 6385 frecabex 6398 fisseneq 6930 mkvprop 7155 exmidfodomrlemr 7200 exmidfodomrlemrALT 7201 acfun 7205 exmidmotap 7259 ccfunen 7262 zfz1isolem1 10815 zfz1iso 10816 uniopn 13392 pw1nct 14634 sbthom 14656 |
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