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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 3173 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
3 | raleqbidva.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | raleqdv 3323 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | bitrd 278 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 |
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 1911 ax-6 1969 ax-7 2009 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-cleq 2722 df-ral 3060 df-rex 3069 |
This theorem is referenced by: raleqbidvv 3327 catpropd 17657 cidpropd 17658 funcpropd 17855 fullpropd 17875 natpropd 17933 gsumpropd2lem 18604 ringurd 20079 istrkgcb 27974 iscgrg 28030 isperp 28230 clwlkclwwlk 29522 urpropd 32648 lindfpropd 32772 opprqus0g 32878 opprqusdrng 32881 ist0cld 33111 matunitlindflem1 36787 sticksstones3 41270 sticksstones11 41278 |
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