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Mirrors > Home > MPE Home > Th. List > ceqsalg | Structured version Visualization version GIF version |
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3380. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.) |
Ref | Expression |
---|---|
ceqsalg.1 | ⊢ Ⅎ𝑥𝜓 |
ceqsalg.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsalg | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsalg.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | ceqsalg.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ax-gen 1869 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
4 | ceqsalt 3377 | . 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) | |
5 | 1, 3, 4 | mp3an12 1561 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1628 = wceq 1630 Ⅎwnf 1855 ∈ wcel 2144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-12 2202 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-v 3351 |
This theorem is referenced by: ceqsal 3381 clel2g 3487 uniiunlem 3839 ralrnmpt2 6921 fimaxre3 11171 pmapglbx 35570 |
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