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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 3174 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
3 | raleqbidva.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | raleqdv 3324 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | bitrd 279 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀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-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-ral 3060 df-rex 3069 |
This theorem is referenced by: raleqbidvv 3332 catpropd 17754 cidpropd 17755 funcpropd 17954 fullpropd 17974 natpropd 18033 gsumpropd2lem 18705 ringurd 20203 istrkgcb 28479 iscgrg 28535 isperp 28735 clwlkclwwlk 30031 urpropd 33222 lindfpropd 33390 opprqus0g 33498 opprqusdrng 33501 ist0cld 33794 matunitlindflem1 37603 primrootsunit1 42079 sticksstones3 42130 |
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