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Mirrors > Home > MPE Home > Th. List > rexeqbidvv | Structured version Visualization version GIF version |
Description: Version of rexeqbidv 3306 with additional disjoint variable conditions, not requiring ax-8 2116 nor df-clel 2812. (Contributed by Wolf Lammen, 25-Sep-2024.) |
Ref | Expression |
---|---|
raleqbidvv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
raleqbidvv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexeqbidvv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbidvv.1 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | raleqbidvv.2 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | notbid 321 | . . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) |
4 | 1, 3 | raleqbidvv 3307 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝜒)) |
5 | ralnex 3150 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓) | |
6 | ralnex 3150 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜒 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜒) | |
7 | 4, 5, 6 | 3bitr3g 316 | . 2 ⊢ (𝜑 → (¬ ∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜒)) |
8 | 7 | con4bid 320 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1542 ∀wral 3054 ∃wrex 3055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-cleq 2731 df-ral 3059 df-rex 3060 |
This theorem is referenced by: rexeqbi1dv 3310 |
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