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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.) |
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 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | ceqsalv.1 | . 2 ⊢ 𝐴 ∈ V | |
3 | ceqsalv.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | ceqsal 3528 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 = wceq 1536 ∈ wcel 2113 Vcvv 3491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1084 df-ex 1780 df-nf 1784 df-cleq 2813 df-clel 2892 |
This theorem is referenced by: ralxpxfr2d 3636 clel4 3653 frsn 5632 raliunxp 5703 idrefALT 5966 funimass4 6723 marypha2lem3 8894 kmlem12 9580 vdwmc2 16310 itg2leub 24330 nmoubi 28547 choc0 29101 nmopub 29683 nmfnleub 29700 elintfv 33028 heibor1lem 35120 elmapintrab 40010 |
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