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Mirrors > Home > MPE Home > Th. List > vtoclegft | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3517.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Jan-2025.) |
Ref | Expression |
---|---|
vtoclegft | ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 262 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜑)) | |
2 | 1 | ax-gen 1798 | . . . . 5 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜑)) |
3 | ceqsalt 3477 | . . . . 5 ⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜑)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑)) | |
4 | 2, 3 | mp3an2 1450 | . . . 4 ⊢ ((Ⅎ𝑥𝜑 ∧ 𝐴 ∈ 𝐵) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑)) |
5 | 4 | ancoms 460 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜑)) |
6 | 5 | biimpd 228 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜑)) |
7 | 6 | 3impia 1118 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-clel 2811 |
This theorem is referenced by: vtoclefex 35855 |
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