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Mirrors > Home > NFE Home > Th. List > vtoclg | GIF version |
Description: Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) |
Ref | Expression |
---|---|
vtoclg.1 | ⊢ (x = A → (φ ↔ ψ)) |
vtoclg.2 | ⊢ φ |
Ref | Expression |
---|---|
vtoclg | ⊢ (A ∈ V → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2489 | . 2 ⊢ ℲxA | |
2 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
3 | vtoclg.1 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
4 | vtoclg.2 | . 2 ⊢ φ | |
5 | 1, 2, 3, 4 | vtoclgf 2913 | 1 ⊢ (A ∈ V → ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 |
This theorem is referenced by: vtoclbg 2915 ceqex 2969 moeq3 3013 mo2icl 3015 sbctt 3108 sbcnestgf 3183 csbing 3462 csbifg 3690 prnzg 3836 sneqrg 3874 unisng 3908 snex 4111 snel1cg 4141 xpkvexg 4285 cnvkexg 4286 p6exg 4290 sikexg 4296 ins2kexg 4305 ins3kexg 4306 iota5 4359 csbiotag 4371 ssfin 4470 csbopabg 4637 vtoclr 4816 csbima12g 4955 dmsnopg 5066 fconstg 5251 fvelimab 5370 fvi 5442 csbovg 5552 trtxp 5781 oqelins4 5794 fnfullfunlem1 5856 fvfullfun 5864 fundmeng 6044 df1c3g 6141 sbthlem2 6204 frecxp 6314 frecxpg 6315 |
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