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Mirrors > Home > MPE Home > Th. List > raleqf | Structured version Visualization version GIF version |
Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
raleq1f.1 | ⊢ Ⅎ𝑥𝐴 |
raleq1f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
raleqf | ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq1f.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | raleq1f.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfeq 2762 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | eleq2 2677 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | imbi1d 330 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜑))) |
6 | 3, 5 | albid 2077 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))) |
7 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
8 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑)) | |
9 | 6, 7, 8 | 3bitr4g 302 | 1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 |
This theorem is referenced by: raleq 3115 raleqbid 3127 dfon2lem3 30934 indexa 32698 ralbi12f 33139 iineq12f 33143 ac6s6f 33151 stoweidlem28 38921 stoweidlem52 38945 fourierdlem31 39031 fourierdlem68 39067 fourierdlem103 39102 fourierdlem104 39103 |
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