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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 3321 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 2917 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | eleq2 2828 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | imbi1d 341 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜑))) |
6 | 3, 5 | albid 2220 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))) |
7 | df-ral 3060 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
8 | df-ral 3060 | . 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 1535 = wceq 1537 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 |
This theorem is referenced by: rexeqf 3352 raleqbid 3354 dfon2lem3 35767 indexa 37720 ralbi12f 38147 iineq12f 38151 ac6s6f 38160 raleqd 45077 stoweidlem28 45984 stoweidlem52 46008 fourierdlem31 46094 fourierdlem68 46130 fourierdlem103 46165 fourierdlem104 46166 |
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