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Mirrors > Home > MPE Home > Th. List > raleqbidva | Structured version Visualization version GIF version |
Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
Ref | Expression |
---|---|
raleqbidva.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
raleqbidva.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
raleqbidva | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbidva.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
2 | 1 | ralbidva 3161 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
3 | raleqbidva.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | raleqdv 3364 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | bitrd 282 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 df-ral 3111 |
This theorem is referenced by: catpropd 16971 cidpropd 16972 funcpropd 17162 fullpropd 17182 natpropd 17238 gsumpropd2lem 17881 istrkgc 26248 istrkgb 26249 istrkgcb 26250 istrkge 26251 iscgrg 26306 isperp 26506 clwlkclwwlk 27787 rngurd 30907 lindfpropd 30996 ist0cld 31186 matunitlindflem1 35053 |
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