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Mirrors > Home > MPE Home > Th. List > r19.29anOLD | Structured version Visualization version GIF version |
Description: Obsolete version of r19.29an 3287 as of 17-Jun-2023. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
r19.29anOLD.1 | ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
r19.29anOLD | ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1914 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nfre1 3305 | . . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜓 | |
3 | 1, 2 | nfan 1899 | . 2 ⊢ Ⅎ𝑥(𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) |
4 | r19.29anOLD.1 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) | |
5 | 4 | adantllr 717 | . 2 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
6 | simpr 487 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 𝜓) | |
7 | 3, 5, 6 | r19.29af 3330 | 1 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ∃wrex 3138 |
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-tru 1539 df-ex 1780 df-nf 1784 df-ral 3142 df-rex 3143 |
This theorem is referenced by: (None) |
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