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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. See raleq 3323 for a version based on fewer axioms. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.) |
| Ref | Expression |
|---|---|
| raleqf.1 | ⊢ Ⅎ𝑥𝐴 |
| raleqf.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| raleqf | ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raleqf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 2 | raleqf.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 3 | 1, 2 | nfeq 2919 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
| 4 | eleq2 2830 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
| 5 | 4 | imbi1d 341 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜑))) |
| 6 | 3, 5 | albid 2222 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))) |
| 7 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
| 8 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑)) | |
| 9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 |
| This theorem is referenced by: rexeqf 3354 raleqbid 3356 dfon2lem3 35786 indexa 37740 ralbi12f 38167 iineq12f 38171 ac6s6f 38180 raleqd 45142 stoweidlem28 46043 stoweidlem52 46067 fourierdlem31 46153 fourierdlem68 46189 fourierdlem103 46224 fourierdlem104 46225 |
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