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Mirrors > Home > MPE Home > Th. List > ralbiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralbi 3166 as of 17-Jun-2023. (Contributed by NM, 6-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralbiOLD | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 3218 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) | |
2 | rspa 3205 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | ralbida 3229 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wral 3137 |
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 1969 ax-7 2014 ax-10 2144 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-ral 3142 |
This theorem is referenced by: (None) |
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