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Mirrors > Home > MPE Home > Th. List > ceqsalv | Structured version Visualization version GIF version |
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid ax-12 2175. (Revised by SN, 8-Sep-2024.) |
Ref | Expression |
---|---|
ceqsalv.1 | ⊢ 𝐴 ∈ V |
ceqsalv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsalv | ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.23v 1940 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)) | |
2 | ceqsalv.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | pm5.74i 271 | . . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)) |
4 | 3 | albii 1816 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)) |
5 | ceqsalv.1 | . . . 4 ⊢ 𝐴 ∈ V | |
6 | 5 | isseti 3496 | . . 3 ⊢ ∃𝑥 𝑥 = 𝐴 |
7 | 6 | a1bi 362 | . 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)) |
8 | 1, 4, 7 | 3bitr4i 303 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-clel 2814 |
This theorem is referenced by: ceqsexv 3530 ralxpxfr2d 3646 frsn 5776 raliunxp 5853 idrefALT 6134 funimass4 6973 fnssintima 7382 imaeqalov 7672 marypha2lem3 9475 kmlem12 10200 vdwmc2 17013 itg2leub 25784 eqscut2 27866 addsuniflem 28049 mulsuniflem 28190 nmoubi 30801 choc0 31355 nmopub 31937 nmfnleub 31954 elintfv 35746 heibor1lem 37796 elmapintrab 43566 |
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