Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ralrexbidOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralrexbid 3241 as of 13-Nov-2023. (Contributed by AV, 21-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralrexbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
ralrexbidOLD | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 3140 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
2 | rspa 3128 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜑) | |
3 | ralrexbid.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜃)) |
5 | 1, 4 | rexbida 3237 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-ral 3066 df-rex 3067 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |