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Mirrors > Home > ILE Home > Th. List > ralsng | GIF version |
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralsng.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralsng | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsns 3630 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | ralsng.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbcieg 2995 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
4 | 1, 3 | bitrd 188 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 [wsbc 2962 {csn 3592 |
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-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-sbc 2963 df-sn 3598 |
This theorem is referenced by: ralsn 3635 ralprg 3643 raltpg 3645 ralunsn 3797 iinxsng 3960 posng 4698 fimax2gtrilemstep 6899 iseqf1olemqk 10491 seq3f1olemstep 10498 fimaxre2 11230 mgm1 12783 sgrp1 12810 mnd1 12841 grp1 12970 0subg 13052 ring1 13229 2sqlem10 14392 nninfsellemdc 14679 |
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