Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > raleqd | Structured version Visualization version GIF version |
Description: Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
raleqd.a | ⊢ Ⅎ𝑥𝐴 |
raleqd.b | ⊢ Ⅎ𝑥𝐵 |
raleqd.e | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
raleqd | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqd.e | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | raleqd.a | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | raleqd.b | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | raleqf 3399 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 Ⅎwnfc 2963 ∀wral 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 |
This theorem is referenced by: allbutfiinf 41701 |
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