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Mirrors > Home > MPE Home > Th. List > rexlimd2 | Structured version Visualization version GIF version |
Description: Version of rexlimd 3276 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
rexlimd2.1 | ⊢ Ⅎ𝑥𝜑 |
rexlimd2.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
rexlimd2.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
rexlimd2 | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimd2.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rexlimd2.3 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
3 | 1, 2 | ralrimi 3180 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
4 | rexlimd2.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
5 | r19.23t 3272 | . . 3 ⊢ (Ⅎ𝑥𝜒 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))) |
7 | 3, 6 | mpbid 235 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-ral 3111 df-rex 3112 |
This theorem is referenced by: rexlimd 3276 sbcrext 3802 |
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