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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elabreximdv | Structured version Visualization version GIF version |
Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) |
Ref | Expression |
---|---|
elabreximdv.1 | ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) |
elabreximdv.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elabreximdv.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) |
Ref | Expression |
---|---|
elabreximdv | ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1957 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1957 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | elabreximdv.1 | . 2 ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓)) | |
4 | elabreximdv.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | elabreximdv.3 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓) | |
6 | 1, 2, 3, 4, 5 | elabreximd 29927 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {cab 2763 ∃wrex 3091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-v 3400 |
This theorem is referenced by: (None) |
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